Introduction to Statistics · Chapter 5 · The Normal Distribution

Normal Distribution (Lap Times)

A lab, not a reading: collect 36 real lap times by a stratified sample, then hold your empirical distribution up against the normal theory that is supposed to describe it.

bookSHelf · Introduction to Statistics · §5.3 · a self-paced section

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Learning objectives — by the end of this section you will be able to

Objectives

Collect a stratified sample of lap times and organize it into a histogram Definitions 5.3.1–5.3.2, §5.3.2
Estimate the theoretical normal distribution that best matches the sample you collected §5.3.3
Calculate the median, quartiles, IQR, percentiles, and a tail probability from your own data §5.3.4
Calculate those same quantities from the theoretical normal model and compare the two sets of answers §5.3.5
Decide whether the normal model is a good description of the data you collected, and defend that decision §5.3.6

5.3

§5.3.1 — Stats Lab: Normal Distribution (Lap Times)

This section is a lab, not a reading. You will collect 36 real lap times by a stratified random sample, build a histogram by hand, and estimate the normal distribution that seems to describe them. Then you will answer the same five questions twice — once from your data, once from the theoretical model — and compare.

The gap between your two sets of answers is not a mistake to explain away. It is the finding: how closely a normal curve traced over your sample actually matches what you counted.

Round relative frequencies and probabilities to four decimal places. Carry all other decimal answers to two places.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

§5.3.1 — naming the two distributions

Empirical Distribution

Definition 5.3.1 — Empirical Distribution

The empirical distribution of a variable is the distribution you get by counting the data you actually collected: the frequency of each value or interval, the quartiles you find by ordering the list, the percentiles you read off positions in that list. It has no formula behind it. Every number in it comes from a value someone wrote down.

Its counterpart is the theoretical distribution, covered next — computed from a model rather than counted from data.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

§5.3.1 — naming the two distributions

Theoretical Distribution

Definition 5.3.2 — Theoretical Distribution

The theoretical distribution is a named probability model — here a normal distribution $X \sim N(\mu, \sigma)$ — chosen to stand in for the process that generated the data. Its median, quartiles, percentiles, and probabilities come from the model's formula, not from any particular data set, so the same model gives the same answers no matter which sample you drew.

Both objects answer the question "what fraction of lap times are below 130 seconds?" — and they will not give you the same number. The empirical answer is a fact about the 36 times you sampled; the theoretical answer is a claim about every lap the racer will ever run, and it is only as good as the normal assumption.

When the two land close together, that assumption is doing honest work. When they land far apart, the gap is the finding — not an error to be explained away.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Insight Note — a tracing over a photograph

The photograph and the tracing

The empirical distribution is the photograph — every bump and gap exactly as your sample happened to fall. The theoretical distribution is the smooth curve someone traced over it. The tracing is easier to work with and it lets you answer questions your photograph never covered, but it is only worth using if it followed the outline closely in the first place. This lab makes you check the tracing against the photograph.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Your turn — empirical or theoretical?

Try It Now 5.3.1

Try It Now 5.3.1 — Hannah's two numbers

Hannah sampled 36 lap times and found that 9 of them came in slower than 131 seconds. Her theoretical model, $N(129.25, 2.50)$ , gives $P(X > 131) = 0.2420$ . Which of these two numbers is the empirical probability and which is the theoretical one, and why are they not equal?

Step 1 — identify the empirical one. It is the one that came from counting. Nine of Hannah's 36 sampled times were slower than 131 seconds, so $P(X > 131) = \dfrac{9}{36} = 0.2500$ — a fact about the 36 values on her worksheet.

Step 2 — identify the theoretical one. The 0.2420 came out of the normal model, by converting 131 to a $z$ -score and reading the area to its right. No lap time was consulted to produce it.

Step 3 — say why they differ. They are answers to different questions. The empirical number describes one sample of 36; draw a different 36 and it will move. The theoretical number describes every lap the model claims she will ever run, and it will not move at all unless you change $\mu$ or $\sigma$ .

Answer: 0.2500 is empirical, 0.2420 is theoretical. They land within a hundredth of each other, which is good news for the normal model — a sample of 36 agreeing that closely with the curve is evidence the curve is describing the process honestly.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

§5.3.2 — the task

Collect the Data

Use a stratified sampling method by lap (races 1 to 20) and a random number generator to pick six lap times from each of laps two through seven — 36 values total (Table 5.3.1).
Construct a histogram by hand: five to six intervals, ruler and pencil, axes scaled after you've counted the tallest interval.
Calculate the sample mean $\bar{x}$ and sample standard deviation $s$ of your 36 values.
Draw a smooth curve through the tops of the bars and describe its shape in one or two sentences.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

§5.3.2 — recording what you collect

Table 5.3.1 — six lap times per lap

Lap 2	Lap 3	Lap 4	Lap 5	Lap 6	Lap 7

Table 5.3.1: Six randomly selected lap times from each of laps two through seven (36 values total).

The strata here are the laps, not the races — every lap gets six of its times into your sample, guaranteeing coverage a purely random draw could not promise.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Try it in rāSHio — loading your 36 lap times

Once the grid above is full, open rāSHio and choose File → Delimited List… to get all 36 of your lap times into the spreadsheet in one paste. Commas, semicolons, tabs, and one-per-line all parse, so you can type the values straight off your paper grid column by column without reformatting anything first.

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Figure 5.3.1

Getting your 36 lap times into rāSHio

File → Delimited List…

Paste or type a list of values with any separator, and the values land in the spreadsheet column ready for analysis.

Figure 5.3.1: Getting your 36 collected lap times into rāSHio: File → Delimited List…

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Try it in rāSHio — drawing which races supply each lap

You need a random number generator to decide WHICH six races supply each lap's times. Open rāSHio, choose File → Random Numbers…, and set Min 1, Max 20 (the twenty races) and How many 6 — then repeat it once per lap, so laps two through seven each get their own independent draw.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Figure 5.3.2

Drawing which six races supply each lap

File → Random Numbers…

Set the minimum, maximum, and how many values to draw, and read off the generated whole numbers — repeat once per lap for an independent draw each time.

Figure 5.3.2: Drawing which six races supply each lap's times: File → Random Numbers…

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Figure 5.3.3

Blank axes for your histogram

Choosing intervals before you draw a single bar

Five or six intervals spread across your smallest and largest lap time is the target. Too few and two fat bars show you nothing; too many and each bar holds one or two values, so the shape is mostly noise. Scale the vertical axis after you've counted the tallest interval, not before.

Figure 5.3.3: Blank axes for sketching your lap-time histogram: lap time across the bottom, frequency up the side.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Try it in rāSHio — checking your sketch

Sketch the histogram by hand first — that is the part of the lab that teaches you what an interval choice does. Then open rāSHio and choose Graph → Histogram on your 36 lap times to check yourself: set the bin start and width to match the intervals you drew, and the bars it returns should be the ones you counted.

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Your turn — mean and standard deviation

Try It Now 5.3.2

Try It Now 5.3.2 — Wen and Mei's stratum

Wen and their wife Mei split the stopwatch duty for one stratum. The six lap times they recorded came out as 127.6, 129.1, 130.4, 128.2, 131.0, and 129.7 seconds. Calculate the sample mean and the sample standard deviation, carrying your answers to two decimal places.

Step 1 — add the six values and divide by six. $\bar{x} = \dfrac{127.6 + 129.1 + 130.4 + 128.2 + 131.0 + 129.7}{6} = \dfrac{776.0}{6} = 129.33$

Step 2 — find how far each value sits from the mean, and square each distance. Working from the unrounded mean of 129.3333: $(-1.7333)^2 = 3.0044$ , $(-0.2333)^2 = 0.0544$ , $(1.0667)^2 = 1.1378$ , $(-1.1333)^2 = 1.2844$ , $(1.6667)^2 = 2.7778$ , $(0.3667)^2 = 0.1344$ .

Step 3 — add the squares, divide by $n-1$ , take the square root. We divide by 5 rather than 6 because this is a sample standard deviation, not a population one: $s = \sqrt{\dfrac{8.3933}{5}} = \sqrt{1.6787} = 1.30$

Answer: $\bar{x} = 129.33$ seconds and $s = 1.30$ seconds. Notice how small that standard deviation is next to the mean — a spread of barely over a second around a lap that takes more than two minutes. A racer that consistent is exactly the kind of process a normal model has a chance of describing well.

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Try it in rāSHio — checking your mean and SD

With your 36 lap times loaded, rāSHio will do the arithmetic of the Try It Now above on the whole sample at once: choose Stats → Summary Statistics and read the sample mean and sample standard deviation straight off the panel. Work $\bar{x}$ and $s$ out by hand first, then use this to check them.

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Figure 5.3.4

Reading your mean and standard deviation

Stats → Summary Statistics

Read the mean, median, standard deviation, quartiles, and IQR for the loaded column straight off the panel.

Figure 5.3.4: Reading the sample mean and standard deviation in rāSHio: Stats → Summary Statistics.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

§5.3.3 — the task

Analyze the Distribution

Using your sample mean, sample standard deviation, and histogram to help, write the approximate theoretical distribution — $X \sim N(\underline{\phantom{xx}}, \underline{\phantom{xx}})$ — then explain how the histogram helped you arrive at it.

A single hump near the middle with the bars falling away at roughly the same rate on both sides is your licence to write $N$ rather than some other letter. The sample mean stands in for $\mu$ and the sample standard deviation stands in for $\sigma$ — estimates, not the real thing, which is exactly why the rest of the lab makes you check them.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Your turn — naming the model

Try It Now 5.3.3

Try It Now 5.3.3 — Ryan's histogram

Ryan's 36 lap times gave $\bar{x} = 129.33$ seconds and $s = 1.30$ seconds. His histogram has one clear hump near 129 seconds, with the bars shrinking at about the same rate on the left and the right. Write the approximate theoretical distribution, and say what in the histogram justifies each part of your answer.

Step 1 — name the family from the shape. One hump, roughly symmetric, tapering on both sides. That is the normal family, so the model is $N$ .

Step 2 — estimate the center. The hump sits near 129 seconds, and the sample mean is 129.33. For a symmetric histogram the mean lands under the peak, so the two agree and $\mu \approx 129.33$ .

Step 3 — estimate the spread. The sample standard deviation is 1.30 seconds, so $\sigma \approx 1.30$ . Sanity-check: about 95% of a normal distribution falls within two standard deviations of the center, so nearly all the bars should sit between $129.33 - 2(1.30) = 126.73$ and $129.33 + 2(1.30) = 131.93$ .

Answer: $X \sim N(129.33, 1.30)$ . The single symmetric hump justifies the normal family, the peak's location justifies the mean, the width of the bars around it justifies the standard deviation. A long tail to one side would have made none of this follow.

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§5.3.4 — the task

Describe the Data

The IQR goes from ___ to ___; $IQR =$ ___ $(IQR = Q_3 - Q_1)$ .
The 15th percentile is ___; the 85th percentile is ___.
The median is ___.
The empirical probability that a randomly chosen lap time is more than 130 seconds is ___.
Explain the meaning of the 85th percentile of this data.

Every one of these comes from counting, not from a formula about $\mu$ and $\sigma$ . Order your 36 times smallest to largest: the median splits the list in half, $Q_1$ is the median of the bottom half, $Q_3$ the median of the top half, and a percentile follows the index rule $i = \dfrac{k}{100}(n+1)$ .

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Your turn — the empirical five-number summary

Try It Now 5.3.4

Try It Now 5.3.4 — Layla and Noor's twelve

Layla and her wife Noor pooled twelve ordered lap times: 125.9, 126.4, 127.1, 127.8, 128.3, 128.9, 129.4, 130.2, 130.8, 131.5, 132.6, 134.1. Find the median, $Q_1$ , $Q_3$ , the IQR, the 15th and 85th percentiles, and the empirical probability that a lap time is more than 130 seconds.

Median (n=12, between the 6th and 7th): $\text{median} = \dfrac{128.9 + 129.4}{2} = 129.15$

Quartiles: $Q_1 = \dfrac{127.1 + 127.8}{2} = 127.45$ , $Q_3 = \dfrac{130.8 + 131.5}{2} = 131.15$ , so $IQR = 131.15 - 127.45 = 3.70$ .

Percentiles (index rule $i = \frac{k}{100}(n+1)$ ): for $P_{15}$ , $i = \frac{15}{100}(13) = 1.95$ , round to positions 1–2: $P_{15} = \dfrac{125.9 + 126.4}{2} = 126.15$ . For $P_{85}$ , $i = \frac{85}{100}(13) = 11.05$ , average positions 11–12: $P_{85} = \dfrac{132.6 + 134.1}{2} = 133.35$ .

Empirical probability: five of twelve times exceed 130 seconds (130.2, 130.8, 131.5, 132.6, 134.1), so $P(X > 130) = \dfrac{5}{12} = 0.4167$ .

Answer: median 129.15, $Q_1 = 127.45$ , $Q_3 = 131.15$ , $IQR = 3.70$ , $P_{15} = 126.15$ , $P_{85} = 133.35$ , $P(X>130)=0.4167$ . The 85th percentile says 85% of these lap times were faster than 133.35 seconds — a lap over 133.35 seconds was one of her worst.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Try it in rāSHio — seeing your quartiles as a picture

The empirical five-number summary is exactly what a box plot draws. Open rāSHio and choose Graph → Box Plot on your 36 lap times to see $Q_1$ , the median, and $Q_3$ marked on the same axis, with the box itself spanning the IQR you just calculated by hand.

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Figure 5.3.5

Seeing your quartiles as a picture

Graph → Box Plot

Calculate, then read the five-number summary off the drawn box and whiskers.

Figure 5.3.5: Seeing your empirical quartiles and IQR as a picture: Graph → Box Plot.

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§5.3.5 — the task

Theoretical Distribution

Answer the same questions again — the IQR endpoints, the IQR, the 15th and 85th percentiles, the median, and $P(X > 130)$ — but this time your ordered list of lap times never gets opened.

Two facts do most of the work. The median equals the mean, because a normal curve is symmetric about $\mu$ . And every percentile is the mean plus some fixed number of standard deviations: $x = \mu + z\sigma$ , where $z$ is whatever score cuts off the area you want.

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§5.3.5 — what the model buys and costs you

The quartiles are fixed by the model

The quartiles are the 25th and 75th percentiles, so $Q_1 = \mu - 0.6745\sigma$ and $Q_3 = \mu + 0.6745\sigma$ — which makes the theoretical IQR exactly $1.349\sigma$ , the same multiple of $\sigma$ for every normal distribution there has ever been.

Once you commit to the normal model, the spacing between the quartiles is no longer something your data gets to decide — only $\sigma$ is. If your empirical IQR disagrees, the model has not made an arithmetic mistake; it is telling you your data's middle half is wider or narrower than a normal curve with that $\sigma$ would produce.

Going the other direction, a probability like $P(X > 130)$ is where the theoretical model outperforms counting — your sample might contain zero laps above 134 seconds, making the empirical probability flatly zero, while the model still assigns it a small positive chance. Neither answer is wrong; they are answers to different questions.

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Context Pause — same questions, different machinery

These are word-for-word the questions you just answered by counting, and that repetition is deliberate. This time your ordered list of lap times never gets opened. Every answer comes out of $N(\mu, \sigma)$ and a $z$ -score, which means you could answer them for a lap the racer has not run yet.

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Your turn — the theoretical five-number summary

Try It Now 5.3.5

Try It Now 5.3.5 — Diego's model

Diego is working from the theoretical model $X \sim N(129.25, 2.50)$ . Help him find the median, $Q_1$ , $Q_3$ , the IQR, the 15th and 85th percentiles, and $P(X > 130)$ . Round the probability to four decimal places and everything else to two.

Median. A normal curve is symmetric about its mean, so the median is the mean: 129.25 seconds.

Quartiles ( $z = \mp0.6745$ ): $Q_1 = 129.25 - 0.6745(2.50) = 127.56$ , $Q_3 = 129.25 + 0.6745(2.50) = 130.94$ , so $IQR = 130.94 - 127.56 = 3.37$ .

Percentiles ( $z = \mp1.0364$ ): $P_{15} = 129.25 - 1.0364(2.50) = 126.66$ , $P_{85} = 129.25 + 1.0364(2.50) = 131.84$ .

Probability. $z = \dfrac{130 - 129.25}{2.50} = 0.30$ , so $P(X > 130) = P(Z > 0.30) = 1 - 0.6179 = 0.3821$ .

Answer: median 129.25, $Q_1 = 127.56$ , $Q_3 = 130.94$ , $IQR = 3.37$ , $P_{15} = 126.66$ , $P_{85} = 131.84$ , $P(X>130) = 0.3821$ . The 85th percentile tells Diego the model predicts 85% of the racer's laps come in under 131.84 seconds — a claim about every lap she will ever run, not about the ones he happened to sample.

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

Try it in rāSHio — theoretical areas and cut-points

Every theoretical answer in this section is an area or a cut-point on one normal curve, and rāSHio will give you both without a table. Choose Distributions → Normal, enter your own $\mu$ and $\sigma$ , then ask it for the area to the right of 130 seconds and for the values at the 15th and 85th percentiles.

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Figure 5.3.6

Computing theoretical percentiles and P(X > 130)

Distributions → Normal

Enter the mean and standard deviation, and read a tail area or an inverse-normal cut-point off the shaded curve.

Figure 5.3.6: Computing the theoretical percentiles and P(X > 130) in rāSHio: Distributions → Normal.

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§5.3.6 — discuss with your group

Do the data from Collect the Data give a close approximation to the theoretical distribution from Analyze the Distribution? Comparing your Describe the Data and Theoretical Distribution answers, explain why or why not.

Line your two sets of numbers up side by side — empirical median against theoretical median, empirical IQR against theoretical IQR — and say how far apart each pair landed. Point at the specific comparison that convinced you, not a general impression.

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Key Terminology

Terms from this lab

empirical distribution — the distribution obtained by counting the data you actually collected; its quartiles, percentiles, and probabilities all come from positions in the ordered sample.

theoretical distribution — a named probability model such as $N(\mu, \sigma)$ used to stand in for the process that generated the data; its quantities come from the model's formula rather than from any one sample.

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The headline result

The gap between your two answers is the finding

Not a rounding error to explain away — evidence for or against the normal model.

A normal model estimated from $\bar{x}$ and $s$ makes one promise: its quartiles sit exactly $1.349\sigma$ apart, no matter whose data produced $\sigma$ . Whether your own 36 lap times deliver a matching IQR is the test the entire lab has been building to.

† A small gap across every comparison — the median, the quartiles, the tail probability — is evidence the normal model is doing honest work; a large one on even a single comparison is evidence it is not, and that finding is just as valid as agreement.

5.3

Normal Distribution (Lap Times) · bookSHelf Intro Stats§5.3

§5.3 — conclusions

What §5.3 leaves you with

The core idea

Empirical numbers come from counting 36 real lap times; theoretical numbers come from a normal model built on $\bar{x}$ and $s$ . They will not agree exactly — the size of the disagreement across every comparison is itself the evidence for whether the normal model fits.

Watch for

The theoretical IQR is pinned at $1.349\sigma$ for every normal distribution — your data doesn't get a vote on that ratio, only on $\sigma$ itself. And an empirical probability can come out a flat zero from a small sample even where the model assigns a real, if small, chance.

Next: §5.4 — Normal Distribution (Pinkie Length), the same lab shape with a variable you can measure on yourself.