1.3 Frequency, Frequency Tables, and Levels of Measurement

Aligned outcomes:

SLO 1

Assess how data were collected and recognize how data collection affects what conclusions can be drawn from the data.

Sorting data as nominal, ordinal, interval, or ratio tells you which arithmetic the measurement will actually support, so before you weigh a conclusion you can ask whether the calculation behind it was legal for that kind of data at all.

SLO 6

Evaluate ethical issues in statistical practice.

Two problems here hand you a published frequency table whose counts do not add up, and a true-sounding claim that misreads a cumulative column as an exact percentage. Checking a table before you quote it is where honest statistical practice begins.

Learning Objectives

By the end of this section, you will be able to:

In this section, you will learn to:
  • round a computed answer to the right number of decimal places for the data you started with;
  • classify a set of data as nominal, ordinal, interval, or ratio, and say which arithmetic that level of measurement allows;
  • build a frequency table with relative frequency and cumulative relative frequency columns from a raw list of data;
  • read percentages and counts off a frequency table, including grouped-interval tables;
  • spot errors in a published frequency table and explain how the mistake changes what the table appears to say.

Once you have a set of data, you need to organize it so that you can see how often each value shows up. That is what this section is about: counting, and then turning those counts into a table you can actually read. Along the way you will run into division, and division produces long decimals — so we start with the simple rule for how far to carry an answer.

1.3.1 Answers and Rounding Off

A simple way to round an answer is to carry your final answer one more decimal place than was present in the original data. Round off only the final answer. Do not round off any intermediate results if you can avoid it. If you do have to round an intermediate result, carry it to at least twice as many decimal places as the final answer.

For example, the average of the three quiz scores four, six, and nine is 6.3, rounded off to the nearest tenth, because the data are whole numbers. Most answers in this course will be rounded off in this manner.

Rounding early is like trimming a board before you measure twice

Every time you round in the middle of a calculation you throw away a sliver of the answer, and those slivers pile up. Keep the full decimal on your calculator until the very last step, then trim once.

It is also not necessary to reduce most fractions in this course. Especially in Probability Topics, the chapter on probability, it is more helpful to leave an answer as an unreduced fraction — the unreduced form usually shows you where the numbers came from.

Here is the rounding rule in action, so you can see how little there is to it. Suppose you are averaging the three quiz scores 4, 6, and 9. The scores are whole numbers, so they carry zero decimal places, which means your final answer carries exactly one. Add them to get 19, divide by 3, and your calculator shows 6.3333333. Do not touch that number until you are finished — if a later step needed it, you would keep every digit the calculator is holding. Only now, at the last step, do you trim to one decimal place: 6.3. Had the scores instead been recorded to one decimal place (4.0, 6.0, 9.0), the same calculation would be reported to two decimal places, as 6.33. The rule never asks you to judge how precise an answer "looks" — it just reads the precision off the data you were given and adds one.

Try It Now 1.3.1

Five students in a study group recorded how many minutes they spent on homework last night. Alex Delgado, who uses they/them pronouns, logged 42 minutes; Hannah Wolcott logged 55; Marcus Bell logged 38; Kayla Nguyen logged 61; and Ethan Shaw logged 47. Find the group's mean study time and round it off correctly.

Solution

Step 1 — Add the values. Alex reported 42, and they are one of five students, so every logged time goes into the sum:

$$ 42 + 55 + 38 + 61 + 47 = 243 $$

Step 2 — Divide by how many values there are:

$$ \frac{243}{5} = 48.6 $$

Step 3 — Round. The original data are whole numbers (zero decimal places), so we carry the final answer to one more decimal place than the data — one decimal place.

Answer: 48.6 minutes. Notice that no rounding was needed here, because the division came out exactly; had it come out 48.62, we would have reported 48.6.

Try it in rāSHio

Open rāSHio, paste the five logged times (42; 55; 38; 61; 47) into File → Delimited List…, then choose Stats → Summary Statistics to read the mean off directly as 48.6 minutes — no adding and dividing by hand. It is worth reopening that panel after the next subsection: a mean is only meaningful once the data reach the interval or ratio level, which is exactly the distinction Levels of Measurement draws.

Figure 1.3.1 — Reading the mean directly in rāSHio: Stats → Summary Statistics.

1.3.2 Levels of Measurement

Definition 1.3.1: Level of Measurement

The level of measurement of a data set is the classification — nominal, ordinal, interval, or ratio — that describes how much mathematical structure the measurements carry, and therefore which calculations are meaningful on them.

The way a set of data is measured is called its level of measurement. Correct statistical procedures depend on a researcher being familiar with levels of measurement, because not every statistical operation can be used with every set of data. Data can be classified into four levels of measurement, listed here from lowest to highest:

Definition 1.3.1 — Measurement-scale ladder Four rounded boxes rise in a staircase from lower-left to upper-right, each box exactly one step higher than the box before it. Every box holds three stacked lines: the level name in bold ink at top, the power that level adds over the one before it in rust accent beneath, and a tiny real-world example in dim grey at the bottom. Left to right: Nominal (named categories; jersey colors), Ordinal (+ order; loyalty tiers), Interval (+ even spacing; temperature degrees F), Ratio (+ true zero; number of machines). The rising boxes are the visual argument -- each scale keeps every property of the one below it and adds exactly one more. Nominal named categories jersey colors Ordinal + order loyalty tiers Interval + even spacing temperature °F Ratio + true zero number of machines

Definition 1.3.1 — The four levels of measurement as a ladder: each step adds one power to the step below it.

Definition 1.3.2: Nominal Scale Level

Data measured on a nominal scale are qualitative (categorical): categories, colors, names, labels, favorite foods, and yes-or-no responses. Nominal scale data are not ordered and cannot be used in calculations.

The level decides what math is legal

You can compute an average shoe size, but not an average favorite pizza topping. Level of measurement is the rule that tells you which of those two situations you are in before you start calculating.

Trying to classify people according to their favorite food does not put anyone ahead of anyone else. Putting pizza first and sushi second is not meaningful — the order is just the order you happened to write them down in.

Smartphone companies are another example of nominal scale data. The data are the names of the companies that make smartphones, and there is no agreed-upon order of these brands, even though people certainly have personal preferences.

Definition 1.3.2 — Nominal scale level Four rounded chips labelled pizza, sushi, tacos and salad sit in a row of four fixed slots. Bold rank tags 1st, 2nd, 3rd and 4th fade in above the slots, as if ranking the chips. The pizza and tacos chips then swap slots along a curved path -- pizza dips below the row, tacos arcs up near the tags -- while the tags stay fixed in place. Next the sushi and salad chips swap the same way. Finally the rank tags fade back out, leaving the chips in their new order: tacos, salad, pizza, sushi. The ranking never meant anything -- nominal categories have no inherent order, so any arrangement is equally valid. 1st 2nd 3rd 4th pizza sushi tacos salad

Definition 1.3.2 — Nominal data are labels: any arrangement of them is equally valid.

Definition 1.3.3: Ordinal Scale Level

Data measured on an ordinal scale are like nominal data in that they are categorical, but with one big difference: ordinal data can be ordered. Differences between two pieces of ordinal data cannot be measured, and ordinal data cannot be used in calculations.

An example of ordinal scale data is a list of the top five national parks in the United States. The parks can be ranked from one to five, but we cannot measure the distance between the park ranked first and the park ranked second.

Another example is a cruise survey where the responses are "excellent," "good," "satisfactory," and "unsatisfactory." These responses are ordered from the most desired response to the least desired — but the gap between "excellent" and "good" is not a measurable quantity, and there is no reason to think it equals the gap between "good" and "satisfactory."

Definition 1.3.3 — Ordinal scale level An order arrow runs left to right beneath four fixed-order chips reading unsatisfactory, satisfactory, good, excellent. A double-headed accent arrow and a question mark sit in each of the three gaps between neighbouring chips. The three gaps breathe in and out independently and out of phase, so their widths are never equal at the same moment and never settle -- only the order of the chips is fixed; the size of each gap is unknowable. ? ? ? unsatisfactory satisfactory good excellent

Definition 1.3.3 — Ordinal data can be ranked, but the size of each gap is unknowable.

Definition 1.3.4: Interval Scale Level

Data measured on an interval scale have a definite ordering and are numerical (quantitative), so differences between values can be calculated. However, an interval scale has no true zero — its zero point is a convention, not an absence of the thing being measured — so ratios are not meaningful.

A true zero means "none of it."

Zero dollars means you have no money. Zero degrees Celsius means the water is freezing — it does not mean there is no temperature. That is the whole difference between an interval scale and a ratio scale, and it is why "80° is twice as hot as 40°" is a sentence that sounds fine and means nothing.

Temperature scales like Celsius (C) and Fahrenheit (F) are measured on the interval scale. In both, 40° is equal to 100° minus 60°, so differences make sense. But 0 degrees does not represent a minimum value: in both scales, 0 is not the absolute lowest temperature. Temperatures like −10° F and −15° C exist, and they are colder than 0.

Definition 1.3.4 — Interval scale level A bare number line running -20 to 100, labelled in degrees Fahrenheit, with a tick every 20 degrees. A curved brace under the label "40deg" spans 20 to 60, then slides right to span 60 to 100 with the same width and the same label, showing that a 40-degree difference is a 40-degree difference no matter where it falls on the scale -- because the scale's zero is a convention, not a true minimum. -20 0 20 40 60 80 100 °F 40°

Definition 1.3.4 — On an interval scale, subtraction works but ratios do not, because zero is a convention.

Definition 1.3.5: Ratio Scale Level

Data measured on the ratio scale give the most information. Ratio scale data are like interval scale data — ordered, numerical, with meaningful differences — but there is a true minimum value of zero, so ratios between values are meaningful.

For example, four multiple-choice statistics final exam scores are 80, 68, 20, and 92 (out of a possible 100 points). The exams are machine-graded.

The data can be put in order from lowest to highest: 20, 68, 80, 92.

The differences between the data have meaning: the score 92 is more than the score 68 by 24 points. Ratios can be calculated because the minimum possible score is 0, so 80 is four times 20 — the score of 80 really is four times better than the score of 20.

Summary of the four levels. The table below collects the four levels introduced above for quick reference.

Summary of the four levels of measurement (nominal, ordinal, interval, ratio) and their properties.
LevelOrdered?Differences meaningful?True zero (ratios meaningful)?Example
NominalNoNoNoCrayon colors, smartphone brands
OrdinalYesNoNoSurvey ratings, park rankings
IntervalYesYesNoTemperature in °C or °F, calendar years
RatioYesYesYesExam scores, distance, income
Definition 1.3.5 — Ratio scale level A vertical Exam-score axis, ticked every 20 from 0 to 100, with a heavy baseline at 0 tagged "true 0". Four bars for scores 20, 68, 80 and 92 rise from that baseline -- 20 and 80 in the accent (rust) tone, 68 and 92 in the curve (blue) tone -- each labelled with its score at the bar top. Dashed guide lines extend right from the tops of the 20 and 80 bars to a curly brace, labelled "80 = 4 x 20" in bold accent text: because the scale has a true zero, 80 really is four times 20. 0 20 40 60 80 100 Exam score true 0 20 68 80 92 80 = 4 × 20

Definition 1.3.5 — A ratio scale is anchored at a true zero, so 80 really is four times 20.

Try It Now 1.3.2

Rachel Whitfield is sorting through the records she kept at her school's field day. Name the level of measurement for each of the following, and say why.

a) The jersey numbers worn by the players on a soccer team.

b) The finishing places in a 5K race: 1st, 2nd, 3rd, …

c) The number of minutes each runner took to finish that same 5K.

Solution

a) Nominal. A jersey number is a label, not a quantity. Player 10 is not "twice" player 5, and averaging jersey numbers produces a number that means nothing.

b) Ordinal. The places are genuinely ordered — 1st beat 2nd — but the gap between 1st and 2nd is not the same measurable amount as the gap between 2nd and 3rd. You cannot subtract places and get a meaningful answer.

c) Ratio. Finishing times are numerical, ordered, differences make sense (one runner finished 4 minutes after another), and zero minutes is a true zero. A 40-minute runner took twice as long as a 20-minute runner.

Answer: a) nominal, b) ordinal, c) ratio. Only the finishing times in part c let Rachel do real arithmetic with her records.

1.3.3 Frequency

Definition 1.3.6: Frequency

A frequency is the number of times a value of the data occurs.

Twenty students were asked how many hours they worked per day. Their responses, in hours, are as follows:

5; 6; 3; 3; 2; 4; 7; 5; 2; 3; 5; 6; 5; 4; 4; 3; 5; 2; 5; 3

Table 1.3.1 lists the different data values in ascending order along with their frequencies.

Table 1.3.1 — Frequency table of student work hours.
Data ValueFrequency
23
35
43
56
62
71

According to Table 1.3.1, there are three students who work two hours, five students who work three hours, and so on. The sum of the values in the frequency column, 20, represents the total number of students included in the sample.

Definition 1.3.6 -- Frequency is a count of how many responses land on each value Twenty response numerals, laid out in the order the definition lists them, sit above six bins for the values 2-7. One value at a time -- 2, then 3, then 4, then 5, then 6, then 7 -- its matching numerals turn accent orange and fly down into that bin, landing as a stack of small squares; the bin's total count fades in above the completed stack. The bins finish at heights 3, 5, 3, 6, 2, 1, matching Table 1.3.1. 2 3 4 5 6 7 5 6 3 3 2 4 7 5 2 3 5 6 5 4 4 3 5 2 5 3 3 5 3 6 2 1

Definition 1.3.6 — Frequency is a count: each dot is one student, and the stack height is the frequency.

Definition 1.3.7: Relative Frequency

A relative frequency is the ratio (fraction or proportion) of the number of times a value of the data occurs in the set of all outcomes to the total number of outcomes.

To find the relative frequencies, divide each frequency by the total number of students in the sample — in this case, 20. Relative frequencies can be written as fractions, percents, or decimals.

Table 1.3.2 — Frequency table of student work hours with relative frequencies.
Data ValueFrequencyRelative Frequency
233/20 or 0.15
355/20 or 0.25
433/20 or 0.15
566/20 or 0.30
622/20 or 0.10
711/20 or 0.05

The sum of the values in the relative frequency column of Table 1.3.2 is \(\frac{20}{20}\), or 1.

Definition 1.3.7 -- Relative frequency turns each count into that value's share of the whole sample Six bars over data values 2-7 are first labelled with their raw counts and count-scaled y-axis ticks. A divide-by-20 badge appears, then the count labels and y-axis ticks relabel as decimal shares of the total -- the bar heights never move, only their meaning changes -- and a "sum = 1.00" readout confirms the shares total one whole. 0 1 2 3 4 5 6 2 3 4 5 6 7 3 5 3 6 2 1 ÷ 20 sum = 1.00

Definition 1.3.7 — Relative frequency turns each count into that value's share of the whole sample.

Definition 1.3.8: Cumulative Relative Frequency

Cumulative relative frequency is the accumulation of the previous relative frequencies. To find the cumulative relative frequencies, add all the previous relative frequencies to the relative frequency for the current row.

A running total, not a fresh count

Think of the cumulative column as a bucket you keep pouring into. Each row adds its own relative frequency to everything already in the bucket, so the last row must come out full — one whole, or 100%.

The cumulative relative frequencies for the student work-hours data are shown in Table 1.3.3.

Table 1.3.3 — Frequency table of student work hours with relative and cumulative relative frequencies.
Data ValueFrequencyRelative FrequencyCumulative Relative Frequency
233/20 or 0.150.15
355/20 or 0.250.15 + 0.25 = 0.40
433/20 or 0.150.40 + 0.15 = 0.55
566/20 or 0.300.55 + 0.30 = 0.85
622/20 or 0.100.85 + 0.10 = 0.95
711/20 or 0.050.95 + 0.05 = 1.00

The last entry of the cumulative relative frequency column is one, indicating that one hundred percent of the data has been accumulated.

Why the column may not land exactly on 1

Because of rounding, the relative frequency column may not always sum to one, and the last entry in the cumulative relative frequency column may not be exactly one. Each should be close to one — if yours is far off, you have an arithmetic error, not a rounding artifact.

Table 1.3.4 represents the heights, in inches, of a sample of 100 semiprofessional soccer players. Notice that the heights have been sorted into intervals rather than listed one value at a time — with 100 measurements of a continuous quantity, almost every player would otherwise get a row of their own.

Table 1.3.4 — Frequency table of soccer player height.
Heights (Inches)FrequencyRelative FrequencyCumulative Relative Frequency
59.95–61.9555/100 = 0.050.05
61.95–63.9533/100 = 0.030.05 + 0.03 = 0.08
63.95–65.951515/100 = 0.150.08 + 0.15 = 0.23
65.95–67.954040/100 = 0.400.23 + 0.40 = 0.63
67.95–69.951717/100 = 0.170.63 + 0.17 = 0.80
69.95–71.951212/100 = 0.120.80 + 0.12 = 0.92
71.95–73.9577/100 = 0.070.92 + 0.07 = 0.99
73.95–75.9511/100 = 0.010.99 + 0.01 = 1.00
Total1001.00

The data in this table have been grouped into the following intervals:

You will meet this table again

This example is used again in Descriptive Statistics, where the method used to compute the intervals will be explained. For now, take the intervals as given and focus on reading the counts off them.

In this sample, there are five players whose heights fall within the interval 59.95–61.95 inches, three players whose heights fall within the interval 61.95–63.95 inches, 15 players whose heights fall within the interval 63.95–65.95 inches, 40 players whose heights fall within the interval 65.95–67.95 inches, 17 players whose heights fall within the interval 67.95–69.95 inches, 12 players whose heights fall within the interval 69.95–71.95 inches, seven players whose heights fall within the interval 71.95–73.95 inches, and one player whose height falls within the interval 73.95–75.95 inches. All heights fall between the endpoints of an interval and not at the endpoints — which is exactly why the boundaries end in .95 rather than on whole inches. No measurement can land on two intervals at once.

Figure 1.3.2 — Frequency histogram of heights for 100 semiprofessional soccer players Eight bars of equal width, touching with no gaps, over the interval boundaries 59.95, 61.95, 63.95, 65.95, 67.95, 69.95, 71.95, 73.95, 75.95. Bar heights (frequency): 5, 3, 15, 40, 17, 12, 7, 1 -- summing to 100 and peaking in the 65.95 to 67.95 interval. Y axis: Frequency, 0 to 45 by 5. X axis: Heights (inches). 59.95 61.95 63.95 65.95 67.95 69.95 71.95 73.95 75.95 0 5 10 15 20 25 30 35 40 45 Heights (inches) Frequency

Figure 1.3.2 — Frequency histogram of the heights of 100 semiprofessional soccer players, in 2-inch intervals.

Definition 1.3.8 — Cumulative relative frequency An axis from 0.00 to 1.00 with a dashed line at 1.00 and x-axis values 2 through 7. Six steps build left to right: each riser rises by that value's relative frequency (0.15, 0.25, 0.15, 0.30, 0.10, 0.05) and its tread draws across the value's slot, with the running cumulative total printed above it. The totals are 0.15, 0.40, 0.55, 0.85, 0.95, 1.00 -- the last step reaches exactly one. 0.00 0.25 0.50 0.75 1.00 2 3 4 5 6 7 0.15 0.40 0.55 0.85 0.95 1.00

Definition 1.3.8 — Cumulative relative frequency is a running total, and the last step must reach one.

Try It Now 1.3.3

Table 1.3.5 shows the amount, in inches, of annual rainfall in a sample of towns.

Table 1.3.5 — Annual rainfall in a sample of 50 towns.
Rainfall (Inches)FrequencyRelative FrequencyCumulative Relative Frequency
2.95–4.9766/50 = 0.120.12
4.97–6.9977/50 = 0.140.12 + 0.14 = 0.26
6.99–9.011515/50 = 0.300.26 + 0.30 = 0.56
9.01–11.0388/50 = 0.160.56 + 0.16 = 0.72
11.03–13.0599/50 = 0.180.72 + 0.18 = 0.90
13.05–15.0755/50 = 0.100.90 + 0.10 = 1.00
Total501.00

From Table 1.3.5, find the percentage of rainfall that is less than 9.01 inches.

Solution

Step 1 — Find the row that ends at 9.01. That is the third row, 6.99–9.01 inches.

Step 2 — Read the cumulative relative frequency for that row. It is 0.56, and the cumulative column already contains every interval below it.

Step 3 — Convert to a percent: \(0.56 = 56\%\).

Answer: 56% of the towns get less than 9.01 inches of rainfall.

Example 1.3.1: Reading a Percentage off the Cumulative Column

From Table 1.3.4, find the percentage of heights that are less than 65.95 inches.

Solution

Step 1 — Identify which rows qualify. If you look at the first, second, and third rows, the heights there are all less than 65.95 inches.

Step 2 — Add the frequencies:

$$ 5 + 3 + 15 = 23 $$

So 23 players are shorter than 65.95 inches.

Step 3 — Convert to a percentage:

$$ \frac{23}{100} = 0.23 = 23\% $$

Answer: 23%. Notice this is exactly the cumulative relative frequency entry in the third row — the cumulative column had already done the adding for us.

Try it in rāSHio

Open rāSHio, paste the twenty students' work-hours list (5; 6; 3; 3; 2; …) into File → Delimited List… — the semicolons parse as-is — then choose Graph → Frequency Table with Discrete values checked to rebuild Table 1.3.1's frequency column along with the relative and cumulative relative frequency columns that follow it.

Figure 1.3.3 — Building Table 1.3.1's frequency columns in rāSHio: Graph → Frequency Table with Discrete values checked.

Try It Now 1.3.4

Daniel Okada and his husband are deciding which town to move to, and he wants to know how common a middling rainfall year is. From Table 1.3.5, find the percentage of rainfall that is between 6.99 and 13.05 inches.

Solution

Step 1 — Identify the rows. Between 6.99 and 13.05 inches covers three rows: 6.99–9.01, 9.01–11.03, and 11.03–13.05.

Step 2 — Add their relative frequencies:

$$ 0.30 + 0.16 + 0.18 = 0.64 $$

Answer: 64% of the towns get between 6.99 and 13.05 inches of rainfall — so most of the towns Daniel and his husband are considering sit in that middle band.

Example 1.3.2: Adding Relative Frequencies for a Middle Band

From Table 1.3.4, find the percentage of heights that fall between 61.95 and 65.95 inches.

Solution

Step 1 — Identify the rows. The band from 61.95 to 65.95 inches is covered by the second row (61.95–63.95) and the third row (63.95–65.95).

Step 2 — Add the relative frequencies in those two rows:

$$ 0.03 + 0.15 = 0.18 $$

Answer: 0.18, or 18%. Because we want a band in the middle rather than everything below a cutoff, we add the individual relative frequencies instead of reading a single cumulative entry.

Try It Now 1.3.5

From Table 1.3.5, find the number of towns that have rainfall between 2.95 and 9.01 inches.

Solution

Step 1 — Identify the rows. The band from 2.95 to 9.01 inches covers the first three rows: 2.95–4.97, 4.97–6.99, and 6.99–9.01.

**Step 2 — The question asks for a number of towns, not a percentage, so add the frequencies rather than the relative frequencies:**

$$ 6 + 7 + 15 = 28 $$

Answer: 28 towns.

Example 1.3.3: Working Every Column of a Grouped Table

Use the heights of the 100 semiprofessional soccer players in Table 1.3.4. Fill in the blanks and check your answers.

a. The percentage of heights that are from 67.95 to 71.95 inches is: ____.

b. The percentage of heights that are from 67.95 to 73.95 inches is: ____.

c. The percentage of heights that are more than 65.95 inches is: ____.

d. The number of players in the sample who are between 61.95 and 71.95 inches tall is: ____.

e. What kind of data are the heights?

f. Describe how you could gather this data (the heights) so that the data are characteristic of all semiprofessional soccer players.

Remember: you count frequencies. To find the relative frequency, divide the frequency by the total number of data values. To find the cumulative relative frequency, add all of the previous relative frequencies to the relative frequency for the current row.

Solution

a. The band 67.95–71.95 covers two rows, with relative frequencies 0.17 and 0.12:

$$ 0.17 + 0.12 = 0.29 = 29\% $$

b. The band 67.95–73.95 covers three rows — add 0.07 to the previous answer:

$$ 0.29 + 0.07 = 0.36 = 36\% $$

c. "More than 65.95 inches" is everything above the third row. The cumulative relative frequency through the third row is 0.23, so everything above it is the rest of the whole:

$$ 1 - 0.23 = 0.77 = 77\% $$

d. Between 61.95 and 71.95 inches covers rows two through six. Add the frequencies this time, because the question asks for a count:

$$ 3 + 15 + 40 + 17 + 12 = 87 $$

e. Height is a measurement that can land anywhere on a continuous scale, so the heights are quantitative continuous data.

f. Get rosters from each team and choose a simple random sample from each. Sampling every team keeps one team's unusual roster from dominating, and randomizing inside each team keeps you from unconsciously picking the tallest players.

Answers: a. 29%, b. 36%, c. 77%, d. 87, e. quantitative continuous, f. get rosters from each team and choose a simple random sample from each.

In your class, have someone conduct a survey of the number of siblings each student has. Create a frequency table. Add to it a relative frequency column and a cumulative relative frequency column. Answer the following questions:

1. What percentage of the students in your class have no siblings?

2. What percentage of the students have from one to three siblings?

3. What percentage of the students have fewer than three siblings?

Try It Now 1.3.6

Table 1.3.5 represents the amount, in inches, of annual rainfall in a sample of towns. What fraction of towns surveyed get between 11.03 and 13.05 inches of rainfall each year?

Solution

Step 1 — Find the row. The interval 11.03–13.05 inches is a single row of Table 1.3.5, with a frequency of 9.

Step 2 — Write it as a fraction of the total. There are 50 towns in the sample.

$$ \frac{9}{50} $$

Answer: \(\frac{9}{50}\), which is 0.18 or 18%.

Example 1.3.4: Finding the Errors in a Published Table

Nineteen people were asked how many miles, to the nearest mile, they commute to work each day. The data are as follows:

2; 5; 7; 3; 2; 10; 18; 15; 20; 7; 10; 18; 5; 12; 13; 12; 4; 5; 10

A student assistant named Claire Bennett compiled the responses into Table 1.3.6:

Table 1.3.6 — Frequency of commuting distances (as published, with errors).
DataFrequencyRelative FrequencyCumulative Relative Frequency
222/190.1053
311/190.1579
411/190.2105
533/190.3684
722/190.4737
1033/190.6316
1222/190.7368
1311/190.7895
1511/190.8421
1811/190.8948
2011/191.0000

a. Is Claire's table correct? If it is not correct, what is wrong?

b. True or False: Three percent of the people surveyed commute three miles or less. If the statement is not correct, what should it be? If the table is incorrect, make the corrections.

c. What fraction of the people surveyed commute five or seven miles?

d. What fraction of the people surveyed commute 12 miles or more? Less than 12 miles? Between five and 13 miles (not including five and 13 miles)?

Solution

a. No, her table is not correct. Add up the frequency column: it sums to 18, not 19, so one person has gone missing. Go back to the raw list and count 18: it appears twice (18; 18), but the table lists a frequency of 1. Because the frequency column is wrong, not all of the cumulative relative frequencies are correct either.

Corrected, the cumulative relative frequency column should read:

$$ \frac{2}{19},\ \frac{3}{19},\ \frac{4}{19},\ \frac{7}{19},\ \frac{9}{19},\ \frac{12}{19},\ \frac{14}{19},\ \frac{15}{19},\ \frac{16}{19},\ \frac{18}{19},\ \frac{19}{19} $$

b. False. The frequency for three miles is one and the frequency for two miles is two, so three people commute three miles or less — that is \(\frac{3}{19} \approx 15.8\%\), not 3%. The "3" in the statement is a count being mistakenly reported as a percent.

c. Five miles has a frequency of 3 and seven miles has a frequency of 2:

$$ \frac{3 + 2}{19} = \frac{5}{19} $$

d. Twelve miles or more: the values 12, 13, 15, 18, 20 have frequencies 2, 1, 1, 2, 1, giving \(\frac{7}{19}\). Less than 12 miles is everything else: \(\frac{12}{19}\). Between five and 13 miles, not including five and 13, means the values 7, 10 and 12, with frequencies 2, 3 and 2: \(\frac{7}{19}\).

Answers: a. No — the frequency column sums to 18 instead of 19, so the cumulative relative frequencies are wrong. b. False; it should be \(\frac{3}{19}\), about 15.8%. c. \(\frac{5}{19}\). d. \(\frac{7}{19}\), \(\frac{12}{19}\), \(\frac{7}{19}\).

Try It Now 1.3.7

Table 1.3.7 contains data for the number of years of service for 70 federal employees.

Table 1.3.7 — Years of service for 70 federal employees.
Number of Years of ServiceNumber of Federal Employees
242
251
263
270
284
296
3011
3112
327
338
346
3510

Answer the following questions.

a. What is the cumulative frequency for years of service between 30 and 35 (inclusive)?

b. What is the relative frequency for 30 years of service?

c. What is the relative frequency for 30 years of service or less?

d. What is the relative frequency for 25 years of service or more?

Solution

a. Add the frequencies for 30 through 35:

$$ 11 + 12 + 7 + 8 + 6 + 10 = 54 $$

b. The frequency for 30 years is 11, out of 70 employees:

$$ \frac{11}{70} \approx 0.157 = 15.7\% $$

c. "30 years or less" is 24 through 30:

$$ \frac{2 + 1 + 3 + 0 + 4 + 6 + 11}{70} = \frac{27}{70} \approx 0.386 = 38.6\% $$

d. "25 years or more" is everything except the two employees with 24 years:

$$ \frac{70 - 2}{70} = \frac{68}{70} \approx 0.971 = 97.1\% $$

Answers: a. 54, b. 11/70 or 0.157 or 15.7%, c. 27/70 or 0.386 or 38.6%, d. 68/70 or 0.971 or 97.1%.

Example 1.3.5: Reading a Two-Column Table That Runs Down the Page

Table 1.3.8 contains the total number of fatal motor vehicle traffic crashes in the United States for a period of 18 years.

Table 1.3.8 — Fatal motor vehicle traffic crashes in the United States over 18 years.
YearTotal Number of CrashesYearTotal Number of Crashes
Year 136,254Year 1138,444
Year 237,241Year 1239,252
Year 337,494Year 1338,648
Year 437,324Year 1437,435
Year 537,107Year 1534,172
Year 637,140Year 1630,862
Year 737,526Year 1730,296
Year 837,862Year 1829,757
Year 938,491Total653,782
Year 1038,477

Answer the following questions.

a. What is the frequency of deaths measured from Year 7 through Year 11?

b. What percentage of deaths occurred after Year 13?

c. What is the relative frequency of deaths that occurred in Year 7 or before?

d. What is the percentage of deaths that occurred in Year 18?

e. What is the cumulative relative frequency for Year 13? Explain what this number tells you about the data.

Solution

a. Add the crash counts for Years 7 through 11:

$$ 37{,}526 + 37{,}862 + 38{,}491 + 38{,}477 + 38{,}444 = 190{,}800 $$

b. "After Year 13" means Years 14 through 18:

$$ 37{,}435 + 34{,}172 + 30{,}862 + 30{,}296 + 29{,}757 = 162{,}522 $$ $$ \frac{162{,}522}{653{,}782} \approx 0.2486 = 24.9\% $$

c. "Year 7 or before" means Years 1 through 7:

$$ 36{,}254 + 37{,}241 + 37{,}494 + 37{,}324 + 37{,}107 + 37{,}140 + 37{,}526 = 260{,}086 $$ $$ \frac{260{,}086}{653{,}782} \approx 0.3978 $$

d. Year 18 alone:

$$ \frac{29{,}757}{653{,}782} \approx 0.0455 = 4.6\% $$

e. The cumulative relative frequency for Year 13 is the running total through Year 13. Years 1 through 13 account for

$$ 653{,}782 - 162{,}522 = 491{,}260 $$

crashes, so

$$ \frac{491{,}260}{653{,}782} \approx 0.7514 $$

This tells you that about 75% of all the fatal crashes in the 18-year period happened in the first 13 years — the crash totals dropped sharply from Year 14 onward, so the later years contribute much less to the total than an even split would suggest.

Answers: a. 190,800, b. about 24.9%, c. about 0.3978, d. about 4.6%, e. about 0.7514 — roughly three-quarters of the period's fatal crashes occurred in the first 13 years.

Problem Set 1.3

Problem 1. What type of measure scale is being used? Nominal, ordinal, interval, or ratio.

a) High school soccer players classified by their athletic ability: Superior, Average, Above average

b) Baking temperatures for various main dishes: 350, 400, 325, 250, 300

c) The colors of crayons in a 24-crayon box

d) Social security numbers

e) Incomes measured in dollars

f) A satisfaction survey of a social website by number: 1 = very satisfied, 2 = somewhat satisfied, 3 = not satisfied

g) Political outlook: extreme left, left-of-center, right-of-center, extreme right

h) Time of day on an analog watch

i) The distance in miles to the closest grocery store

j) The dates 1066, 1492, 1644, 1947, and 1944

k) The heights of 21–65 year-old women

l) Common letter grades: A, B, C, D, and F

Solution

Step 1 — Ask the three sorting questions for each item: Can the values be ordered? If yes, are the differences between them measurable? If yes, is there a true zero so that ratios make sense?

Step 2 — Apply them item by item:

a) Superior / Average / Above average are ranked labels — ordered, but the gap between them is not measurable. Ordinal.

b) Baking temperatures are numerical with meaningful differences, but 0° is not "no heat," so ratios are meaningless. Interval.

c) Crayon colors are pure labels with no order. Nominal.

d) Social security numbers look numerical but are identifiers; averaging them is meaningless. Nominal.

e) Income in dollars is ordered, differences are meaningful, and $0 is a true zero. Ratio.

f) The codes 1 = very satisfied, 2 = somewhat satisfied, 3 = not satisfied are ranked categories; the numbers are stand-ins for words. Ordinal.

g) Extreme left / left-of-center / right-of-center / extreme right are political categories with no agreed numerical spacing, and the "order" is a spectrum of kind rather than of amount. Nominal.

h) Clock time on an analog watch has meaningful differences (3:00 to 5:00 is two hours), but 12:00 is a convention, not an absence of time. Interval.

i) Distance in miles has a true zero and meaningful ratios. Ratio.

j) Calendar dates are ordered with meaningful differences, but year 0 is a convention, not the beginning of time. Interval.

k) Height has a true zero and meaningful ratios. Ratio.

l) Letter grades A, B, C, D, F are ranked, but the gap from A to B is not a measured quantity. Ordinal.

Answer: a) ordinal, b) interval, c) nominal, d) nominal, e) ratio, f) ordinal, g) nominal, h) interval, i) ratio, j) interval, k) ratio, l) ordinal.

Problem 2. Mateo Alvarez surveyed fifty part-time students at his college, asking how many courses they were taking this term. His (incomplete) results are shown in Table 1.3.9.

Table 1.3.9 — Part-time student course loads.
# of CoursesFrequencyRelative FrequencyCumulative Relative Frequency
1300.6
215
3

a) Fill in the blanks in Table 1.3.9.

b) What percent of students take exactly two courses?

c) What percent of students take one or two courses?

Solution

Step 1 — Use the total to recover the missing frequency. Fifty students were surveyed, and 30 plus 15 are already accounted for:

$$ 50 - 30 - 15 = 5 $$

So 5 students take three courses.

Step 2 — Divide each frequency by 50 to get relative frequencies:

$$ \frac{30}{50} = 0.6, \qquad \frac{15}{50} = 0.3, \qquad \frac{5}{50} = 0.1 $$

Step 3 — Accumulate down the column — each row adds its own relative frequency to the running total:

$$ 0.6, \qquad 0.6 + 0.3 = 0.9, \qquad 0.9 + 0.1 = 1.0 $$

The completed table is:

Completed Table 1.3.9 — part-time student course loads, with all frequencies filled in.
# of Courses Frequency Relative Frequency Cumulative Relative Frequency
1 30 0.6 0.6
2 15 0.3 0.9
3 5 0.1 1.0

Step 4 — Read the two percentages off the finished table. "Exactly two courses" is the relative frequency of the second row; "one or two courses" is the cumulative relative frequency through the second row.

Answer: a) 5 students take three courses; relative frequencies 0.6, 0.3, 0.1; cumulative relative frequencies 0.6, 0.9, 1.0. b) 30% take exactly two courses. c) 90% take one or two courses.

Problem 3. Sixty adults with gum disease were asked the number of times per week they used to floss before their diagnosis. The (incomplete) results are shown in Table 1.3.10.

Table 1.3.10 — Flossing frequency for adults with gum disease.
# Flossing per WeekFrequencyRelative FrequencyCumulative Relative Freq.
0270.4500
118
30.9333
630.0500
710.0167

a) Fill in the blanks in Table 1.3.10.

b) What percent of adults flossed six times per week?

c) What percent flossed at most three times per week?

Solution

Step 1 — Fill in the relative frequency for one flossing per week. There are 60 adults in the study:

$$ \frac{18}{60} = 0.3000 $$

Step 2 — Accumulate down to the row you can see. The cumulative relative frequency for 0 times is 0.4500 and for 1 time is \(0.4500 + 0.3000 = 0.7500\). The table already tells us the cumulative value at 3 times is 0.9333, so the relative frequency of that row is the difference:

$$ 0.9333 - 0.7500 = 0.1833 $$

Step 3 — Turn that relative frequency back into a count:

$$ 0.1833 \times 60 \approx 11 $$

Step 4 — Finish the cumulative column:

$$ 0.9333 + 0.0500 = 0.9833, \qquad 0.9833 + 0.0167 = 1.0000 $$

The completed table is:

Completed Table 1.3.10 — flossing frequency for adults with gum disease, with all frequencies filled in.
# Flossing per Week Frequency Relative Frequency Cumulative Relative Freq.
0 27 0.4500 0.4500
1 18 0.3000 0.7500
3 11 0.1833 0.9333
6 3 0.0500 0.9833
7 1 0.0167 1.0000

Step 5 — Read off the two percentages. "Six times per week" is a single row's relative frequency; "at most three times" is the cumulative relative frequency through the 3-times row.

Answer: a) The 3-times row has frequency 11 and relative frequency 0.1833; the cumulative column reads 0.4500, 0.7500, 0.9333, 0.9833, 1.0000. b) 5.00% flossed six times per week. c) 93.33% flossed at most three times per week.

Problem 4. Nineteen immigrants to the U.S. were asked how many years, to the nearest year, they have lived in the U.S. The data are as follows: 2; 5; 7; 2; 2; 10; 20; 15; 0; 7; 0; 20; 5; 12; 15; 12; 4; 5; 10. Table 1.3.11 was produced.

Table 1.3.11 — Frequency of immigrant survey responses (as published).
DataFrequencyRelative FrequencyCumulative Relative Frequency
022/190.1053
233/190.2632
411/190.3158
533/190.4737
722/190.5789
1022/190.6842
1222/190.7895
1511/190.8421
2011/191.0000

a) Fix the errors in Table 1.3.11. Also, explain how someone might have arrived at the incorrect number(s).

b) Explain what is wrong with this statement: "47 percent of the people surveyed have lived in the U.S. for 5 years."

c) Fix the statement in b to make it correct.

d) What fraction of the people surveyed have lived in the U.S. five or seven years?

e) What fraction of the people surveyed have lived in the U.S. at most 12 years?

f) What fraction of the people surveyed have lived in the U.S. fewer than 12 years?

g) What fraction of the people surveyed have lived in the U.S. from five to 20 years, inclusive?

Solution

Step 1 — Recount the raw data. Sort the 19 responses: 0; 0; 2; 2; 2; 4; 5; 5; 5; 7; 7; 10; 10; 12; 12; 15; 15; 20; 20. The true frequencies are 0 → 2, 2 → 3, 4 → 1, 5 → 3, 7 → 2, 10 → 2, 12 → 2, 15 → 2, 20 → 2, and these sum to 19.

Step 2 — Compare with the published table. The table lists 15 → 1 and 20 → 1. Both are one too low, and the published frequency column therefore sums to 17, not 19. Because 15's frequency is wrong, its cumulative entry is wrong too: 0.8421 corresponds to \(\frac{16}{19}\) when it should be \(\frac{17}{19} \approx 0.8947\).

Step 3 — How the mistake likely happened. The raw list is unsorted, and both 15 and 20 appear twice, far apart in the list. Someone tallying by scanning left-to-right without sorting first is exactly the person who catches the first 15 and misses the second. Sorting the data before tallying prevents it, and summing the frequency column against the sample size catches it.

The corrected table:

Completed frequency table for years lived in the U.S., with all frequencies filled in.
Data Frequency Relative Frequency Cumulative Relative Frequency
0 2 2/19 0.1053
2 3 3/19 0.2632
4 1 1/19 0.3158
5 3 3/19 0.4737
7 2 2/19 0.5789
10 2 2/19 0.6842
12 2 2/19 0.7895
15 2 2/19 0.8947
20 2 2/19 1.0000

Step 4 — Diagnose the statement in part b. The value 0.4737 sits in the cumulative column, so it describes everyone at five years or fewer, not everyone at exactly five years. Exactly five years is the relative frequency \(\frac{3}{19} \approx 15.8\%\). The statement also rounds 47.37% down to "47 percent" while presenting it as exact.

Step 5 — Read the remaining fractions off the corrected frequencies.

Five or seven years: \(3 + 2 = 5\), so \(\frac{5}{19}\).

At most 12 years: \(2 + 3 + 1 + 3 + 2 + 2 + 2 = 15\), so \(\frac{15}{19}\).

Fewer than 12 years: drop the 12s, \(15 - 2 = 13\), so \(\frac{13}{19}\).

Five to 20 years inclusive: \(3 + 2 + 2 + 2 + 2 + 2 = 13\), so \(\frac{13}{19}\).

Answer: a) The frequencies for 15 and 20 should each be 2, not 1, and the cumulative entry for 15 should be \(\frac{17}{19} \approx 0.8947\); the error comes from tallying an unsorted list and missing the second occurrence of each repeated value. b) 0.4737 is a cumulative figure, so it describes people who have lived in the U.S. five years or less, not exactly five years. c) "About 47.4 percent of the people surveyed have lived in the U.S. for five years or less." d) \(\frac{5}{19}\). e) \(\frac{15}{19}\). f) \(\frac{13}{19}\). g) \(\frac{13}{19}\).

Problem 5. How much time does it take to travel to work? Table 1.3.12 shows the mean commute time by state for workers at least 16 years old who are not working at home. Find the mean travel time, and round off the answer properly.

Table 1.3.12 — Mean commute time by state, in minutes.
24.024.325.918.927.517.921.820.916.727.3
18.224.720.022.623.918.031.422.324.025.5
24.724.628.124.922.623.623.425.724.825.5
21.225.723.123.023.926.016.323.121.421.5
27.027.018.631.723.330.122.923.321.718.6
Solution

Step 1 — Add all 50 state commute times. Adding row by row gives row totals of 225.2, 230.6, 247.9, 225.2, and 244.2:

$$ 225.2 + 230.6 + 247.9 + 225.2 + 244.2 = 1{,}173.1 $$

Step 2 — Divide by the number of states to get the mean:

$$ \frac{1{,}173.1}{50} = 23.462 $$

Step 3 — Round properly. Each state's travel time was measured to the nearest tenth (one decimal place), so the final answer carries one more decimal place than the data — two decimal places. Round 23.462 to the nearest hundredth.

Answer: The mean travel time is 23.46 minutes.

Problem 6. A leading business magazine publishes data on small businesses (defined as businesses that have been publicly traded for at least a year, have a stock price of at least $5 per share, and have reported annual revenue between $5 million and $1 billion). Table 1.3.13 shows the ages of the chief executive officers for the first 60 ranked small businesses.

Table 1.3.13 — Ages of chief executive officers at the first 60 ranked small businesses.
AgeFrequencyRelative FrequencyCumulative Relative Frequency
40–443
45–4911
50–5413
55–5916
60–6410
65–696
70–741

a) What is the frequency for CEO ages between 54 and 65?

b) What percentage of CEOs are 65 years or older?

c) What is the relative frequency of ages under 50?

d) What is the cumulative relative frequency for CEOs younger than 55?

e) Which graph shows the relative frequency and which shows the cumulative relative frequency?

Figure 1.3.4 — CEO ages, Graph A A bar chart with seven bars over the age brackets 40-44 through 70-74. Bars rise from the shortest at 40-44, peak at 55-59, then fall back down through 70-74 -- a single-peaked, roughly symmetric shape. The y-axis is labelled Relative Frequency and runs from 0.0 to 1.0; the chart itself is not otherwise titled with what it measures. Graph A 0.0 0.2 0.4 0.6 0.8 1.0 40–44 45–49 50–54 55–59 60–64 65–69 70–74 Relative Frequency CEO's Ages

Figure 1.3.4 — Graph A of the CEO age data from Table 1.3.13.

Figure 1.3.5 -- Graph B, cumulative relative frequency of CEO ages Seven bars over brackets 40-44 through 70-74 on the x-axis, labelled CEO's Ages. The y-axis is labelled Relative Frequency with gridlines at 0.0 through 1.0 in steps of 0.2. Bar heights are the running total of the relative frequencies, so they never fall and the last bar reaches exactly 1.0. Graph B 0.0 0.2 0.4 0.6 0.8 1.0 40–44 45–49 50–54 55–59 60–64 65–69 70–74 Relative Frequency CEO's Ages

Figure 1.3.5 — Graph B of the CEO age data from Table 1.3.13.

Solution

Step 1 — Confirm the sample size. The frequencies add to \(3 + 11 + 13 + 16 + 10 + 6 + 1 = 60\), which matches the 60 ranked small businesses.

Step 2 — Part a: frequency for ages between 54 and 65. Those ages are covered by the 55–59 and 60–64 intervals:

$$ 16 + 10 = 26 $$

Step 3 — Part b: percentage 65 or older. That is the 65–69 and 70–74 intervals:

$$ \frac{6 + 1}{60} = \frac{7}{60} \approx 0.1167 = 11.67\% $$

Step 4 — Part c: relative frequency of ages under 50. That is the 40–44 and 45–49 intervals:

$$ \frac{3 + 11}{60} = \frac{14}{60} \approx 0.2333 $$

Step 5 — Part d: cumulative relative frequency for CEOs younger than 55. Accumulate the first three intervals:

$$ \frac{3 + 11 + 13}{60} = \frac{27}{60} = 0.45 $$

Step 6 — Part e: tell the two graphs apart by their shape. A relative-frequency graph shows each interval on its own, so its bars rise and then fall with the data (here, peaking at 55–59). A cumulative graph can never go down — each bar includes everything before it — so its bars climb steadily and the last bar reaches 1. Graph A rises to a peak at 55–59 and then falls away; Graph B climbs without ever dropping and tops out at 1.

Answer: a) 26. b) About 11.67%. c) About 0.2333. d) 0.45. e) Graph A shows the relative frequency; Graph B shows the cumulative relative frequency.

Problem 7. Table 1.3.14 contains data on hurricanes that have made direct hits on the U.S. between 1851 and 2004. A hurricane is given a strength category rating based on the minimum wind speed generated by the storm.

Table 1.3.14 — Frequency of hurricane direct hits.
CategoryNumber of Direct HitsRelative FrequencyCumulative Frequency
11090.39930.3993
2720.26370.6630
3710.2601
4180.9890
530.01101.0000
Total273

What is the relative frequency of direct hits that were category 4 hurricanes?

a) 0.0768

b) 0.0659

c) 0.2601

d) Not enough information to calculate

Solution

Step 1 — Find the count for category 4 and the total. Table 1.3.14 lists 18 direct hits for category 4, out of 273 direct hits in all.

Step 2 — Divide to get the relative frequency:

$$ \frac{18}{273} \approx 0.0659 $$

Note that the table's blank cell for this row is exactly what you just computed — the data needed were all present, so choice d is wrong.

Answer: b) 0.0659.

Problem 8. Using Table 1.3.14 again: what is the relative frequency of direct hits that were AT MOST a category 3 storm?

a) 0.3480

b) 0.9231

c) 0.2601

d) 0.3370

Solution

Step 1 — Translate "at most a category 3." That means category 1, 2, or 3 — every storm up to and including category 3.

Step 2 — Add those direct hits:

$$ 109 + 72 + 71 = 252 $$

Step 3 — Divide by the total number of direct hits:

$$ \frac{252}{273} \approx 0.9231 $$

Alternatively, subtract the categories above 3 from the whole: \(1 - \frac{18 + 3}{273} = 1 - 0.0769 = 0.9231\).

Answer: b) 0.9231.

Key Terms

level of measurement — the classification (nominal, ordinal, interval, or ratio) describing how much mathematical structure a data set carries, and therefore which calculations are meaningful.

nominal scale level — categorical data that are labels only; not ordered and not usable in calculations.

ordinal scale level — categorical data that can be ranked, but whose differences cannot be measured.

interval scale level — numerical data that are ordered with meaningful differences, but no true zero, so ratios are meaningless.

ratio scale level — numerical data that are ordered, with meaningful differences and a true zero, so ratios are meaningful.

frequency — the number of times a value of the data occurs.

relative frequency — the ratio of the number of times a value occurs to the total number of outcomes.

cumulative relative frequency — the running total of the relative frequencies up to and including the current row.