Introduction to Statistics · Chapter 2 · Descriptive Statistics
The mean, the median, and the mode — three different answers to "what is typical here" — and the arithmetic that turns a raw list, or a frequency table, into each one.
bookSHelf · Introduction to Statistics · §2.5 · a self-paced section
Learning objectives — by the end of this section you will be able to
§2.5.1 — two ways to describe the middle
The mean is a balancing point: add every value and share the total out evenly. The median is a location: order the data and find the value standing at the middle address. They usually land close together — and when they don't, that gap is the story.
Same arithmetic, done two ways, always agrees.
§2.5.1 — the balancing point
Definition 2.5.1 — Mean (Arithmetic Mean)
The mean of a data set is the sum of all the data values divided by the number of data values. A sample mean is written x ("x-bar"); a population mean is written μ ("mew"). For x to be a good estimate of μ, the sample must be taken truly at random.
Figure: the mean is the level every bar reaches once the total is shared out evenly.
§2.5.1 — two ways to do the arithmetic, always agree
The sample — eleven values
1; 1; 1; 2; 2; 3; 4; 4; 4; 4; 4
Add every value once
x=111+1+1+2+2+3+4+4+4+4+4=2.7
Multiply by frequency
Values 1, 2, 3, 4 occur 3, 2, 1, 5 times:
x=11(1)(3)+(2)(2)+(3)(1)+(4)(5)=1130=2.7
Same number, less writing. That frequency-weighted form is worth holding onto — it is the seed of the grouped-data formula that closes this section.
§2.5.1 — an address, then a value
Definition 2.5.2 — Median
The median is the value that splits ordered data into two halves. Its location in the ordered list is
2n+1
If n is odd, the median is the single middle value. If n is even, average the two middle values. Often written M.
Figure: the location formula gives an address — the median is the value standing at it.
Insight Note — the location is an address, not a value
The location is an address, not a value.
Asking "where is the median?" is like asking which seat in a sold-out row is the middle seat. Seat 21 is the address; the person sitting in it is the answer. Confusing the two is the single most common mistake in this section.
§2.5.1 — the address, twice
97 values (odd)
2n+1=297+1=49
The median is the 49th value — one single position.
100 values (even)
2n+1=2100+1=50.5
The median sits midway between the 50th and 51st — average those two.
49 and 50.5 are addresses: they tell you where to look, not what you will find there. The median is usually the better center when the data has extreme values — it only counts how many values sit on each side, never how far away.
Context Pause — "mean" and "average" are the same word here
"Mean" and "average" are the same word here.
In everyday speech people swap the two freely. The technical term is the arithmetic mean, and "average" is really a loose word for any center — but among non-statisticians "average" is universally accepted for the arithmetic mean, and this book uses both.
Your turn — 39 waiting times
Try It Now 2.5.1 — months on the transplant list
3; 4; 5; 7; 7; 7; 7; 8; 8; 9; 9; 10; 10; 10; 10; 10; 11; 12; 12; 13; 14; 14; 15; 15; 17; 17; 18; 19; 19; 19; 21; 21; 22; 22; 23; 24; 24; 24; 24. Calculate the mean and the median.
x≈13.9 months, M=13 months
39 values sum to 544, so x=544/39≈13.9. With n=39, 2n+1=20 — the median is the 20th value exactly (39 is odd), which is 13.
Worked example — mean and median, side by side
Example 2.5.1 — forty ER patient ages (years)
3; 4; 8; 8; 10; 11; 12; 13; 14; 15; 15; 16; 16; 17; 17; 18; 21; 22; 22; 24; 24; 25; 26; 26; 27; 27; 29; 29; 31; 32; 33; 33; 34; 34; 35; 37; 40; 44; 44; 47. Calculate the mean and the median.
Forty values. Group the repeats to save writing before you add — the frequency form from Definition 2.5.1 is exactly this shortcut.
Example 2.5.1 — worked through
Mean
x=40943=23.6
Median
n=40, so 2n+1=20.5 — average the 20th and 21st values, both 24:
M=224+24=24
x=23.6 and M=24 — the two agree closely, which is what you expect when no single value is wildly out of line with the rest.
Try it in rāSHio
Paste the forty emergency-room ages into File → Delimited List… — the semicolons parse as-is — then choose Stats → Summary Statistics. The mean 23.6 and the median 24 come back side by side.
That side-by-side comparison is exactly what the rest of this section is about.
Figure 2.5.1 — Stats → Summary Statistics
Load the column, open Summary Statistics, and the mean, median, and mode all arrive at once.
Use it to check work done by hand — the tool cannot tell you which measure is the honest one for your data, only compute all three.
Figure 2.5.1: reading the mean and the median in rāSHio.
§2.5.2 — when one extreme value changes everything
So far the mean and the median have stayed close. That is the normal case. The reason to keep both around is what happens when a single value sits very far from the rest.
The mean drags toward an extreme; the median just counts it as one more value.
Context Pause — this is why "average income" is a loaded phrase
This is why "average income" is a loaded phrase.
When a news report quotes an average income, salary, or house price, ask which center they used. A handful of very large incomes can push a mean far above what a typical household actually earns — the median is usually the honest answer to "what does a normal person make?"
Your turn — one mansion among sixty houses
Try It Now 2.5.2 — sixty households
One house is worth $2,500,000. Twenty-nine houses are worth $280,000, and all the others are worth $315,000. Which is the better measure of the center: the mean or the median?
Median = $315,000 is the better measure
x=602,500,000+(280,000)(29)+(315,000)(30)=334,500 The mean, $334,500, sits above 59 of the 60 houses. The median location 261=30.5 averages the 30th and 31st values — both $315,000.
Worked example — one salary, forty-nine others
Example 2.5.2 — a town of 50 people
One person earns $5,000,000 per year; the other 49 each earn $30,000. Which is the better measure of the "center" for a newspaper article: the mean or the median?
A local newspaper is describing what a typical household there earns — that framing is the whole question.
Example 2.5.2 — worked through
Mean
x=50(30,000)(49)+5,000,000=129,400
Median
250+1=25.5 — the 25th and 26th values are both $30,000, so M=$30,000.
Not one person in this town earns $129,400 — 49 of the 50 earn exactly $30,000. The median is the better measure of the center; the $5,000,000 is an outlier.
§2.5.3 — a third question entirely
The mean asks for a balancing point; the median asks for a splitting location. The mode asks a different question altogether: what value actually happens most often?
The only measure of center that is always a real member of the data.
§2.5.3 — what occurs most often
Definition 2.5.3 — Mode
The mode of a data set is the value that occurs most frequently. A data set can have more than one mode, as long as those values share the same — and the highest — frequency.
Figure: the mode is whichever score's stack finishes tallest.
§2.5.3 — a tie for tallest
Definition 2.5.4 — Bimodal
A data set with exactly two modes is called bimodal.
Figure: two scores tie for the highest frequency, so the mode label lands twice.
Unlike the mean and the median, the mode is always an actual member of the data set — a value someone really recorded, not a computed balancing point. That is both its strength and its limitation.
Your turn — books checked out
Try It Now 2.5.3 — 25 students
0; 0; 0; 1; 2; 3; 3; 4; 4; 5; 5; 7; 7; 7; 7; 8; 8; 8; 9; 10; 10; 11; 11; 12; 12. Find the mode.
Mode = 7 books
0 appears three times; 3, 4, 5, 10, 11, 12 each appear twice; 8 appears three times; 7 appears four times — the highest tally, so 7 wins outright.
Worked example — the most common exam score
Example 2.5.3 — 20 students
50; 53; 59; 59; 63; 63; 72; 72; 72; 72; 72; 76; 78; 81; 83; 84; 84; 84; 90; 93. Find the mode.
Work left to right and tally as you go — the count that wins is the only thing that matters.
Example 2.5.3 — worked through
Mode = 72
59 appears twice, 63 appears twice, 72 appears five times, 84 appears three times, and every other score appears once. Five is the largest frequency, and it belongs to 72.
Try it in rāSHio
Paste the twenty exam scores into File → Delimited List…, then choose Graph → Frequency Table with Discrete values checked. Every distinct score gets its own row with its count beside it.
The mode stops being something you hunt for and becomes the row with the largest number in it — 72, at five.
Figure 2.5.2 — Graph → Frequency Table
One row per distinct value, with its count printed beside it — the mode is whichever row has the largest number.
The same table also carries the relative and cumulative frequency columns §2.3 used for percentiles.
Figure 2.5.2: counting each distinct value in rāSHio.
§2.5.4 — when typical beats balanced
The mode earns its keep whenever the typical value matters more than the balancing point — especially when the data clusters in two places instead of one, a fact neither the mean nor the median will report.
The mode is the only measure of center that also works on categories, not just numbers.
Your turn — two questions, one idea
Try It Now 2.5.4 — credit scores, then factory earnings
Five credit scores are 680, 680, 700, 720, 720 — explain why this is bimodal. A factory's mode is $25,000 (150 of 301 workers), median $50,000, mean $47,500 — which is the best center?
Bimodal, by 680 and 720
680 occurs twice, 720 occurs twice — two values tie for the highest frequency.
Mode, $25,000
150 of 301 workers — nearly half — earn exactly $25,000. The median and mean both sit near double that, describing no one's actual paycheck.
Worked example — two ties, one label
Example 2.5.4 — five real estate exam scores
430, 430, 480, 480, 495. Explain why this data set is bimodal.
Bimodal: modes 430 and 480
430 occurs twice, 480 occurs twice, 495 occurs once. Two different values tie for the highest frequency, so both are modes.
Context Pause — the mode works on words, too
The mode works on words, too.
The mean and the median need numbers, but the mode only needs counting. If the data is red, red, red, green, green, yellow, purple, black, blue, the mode is red — a perfectly good answer to "what color are these?" This makes the mode the only one of the three measures that also works on categorical data.
§2.5.5 — why a sample mean means anything at all
A single sample mean is only useful as an estimate of something real if bigger samples reliably get closer to that real number. That promise has a name.
Very likely, not guaranteed — that word choice matters.
§2.5.5 — bigger samples, closer means
Definition 2.5.5 — Law of Large Numbers
As the sample size drawn from a population increases, the sample mean x becomes very likely to be closer and closer to the population mean μ.
Figure: as the sample size grows, the sample mean settles toward the population mean.
This is the reason a single sample mean is treated as an estimate of something real, rather than as an accident of whoever happened to get surveyed.
Insight Note — bigger samples buy you accuracy, not certainty
Bigger samples buy you accuracy, not certainty.
The law says "very likely," not "guaranteed." A large sample can still miss — it is just far less likely to miss by much. Think of it as the noise averaging out rather than the truth being revealed.
Your turn — 30 voters or 3,000?
Try It Now 2.5.5 — a polling budget
A pollster's budget covers surveying either 30 or 3,000 voters from the same population. Which sample's mean is more likely to land close to μ, and what result says so?
The 3,000-voter sample — by the Law of Large Numbers
3,000 is far larger than 30 from the same population, so its sample mean is more likely to be close to μ. This is a statement about likelihood — the small sample could still get lucky, it is just much less dependable.
§2.5.6 — naming what a sample gives you
Every number you have computed so far — the mean, the median, the mode — is a statistic: something calculated from a sample, standing in for a population number you cannot reach directly.
You cannot measure μ, so you compute x and use it as your best stand-in.
§2.5.6 — a number computed from a sample
Definition 2.5.6 — Statistic
A statistic is a number calculated from a sample. The mean, the median, and the mode are all statistics. x is the statistic used to estimate the population mean μ.
Figure: a statistic stands in for a population number nobody can reach.
A sampling distribution is a relative frequency distribution built from a great many samples — the object the next slide's table is a small, thirty-sample preview of.
§2.5.6 — thirty samples, one preview
| # of movies | Rel. freq. |
|---|---|
| 0 | 5/30 |
| 1 | 15/30 |
| 2 | 6/30 |
| 3 | 3/30 |
| 4 | 1/30 |
Table 2.5.1: movies watched by 30 randomly selected students.
If you let the number of samples get very large — say, 300 million or more — this relative frequency table becomes a relative frequency distribution: the same idea, at the scale where it earns the name.
Try it in rāSHio
Enter the thirty students' movie counts, then choose Graph → Frequency Table with Discrete values checked. The relative frequency column Table 2.5.1 prints as fifths and halves of thirty is built for you.
That column is exactly what turns into a sampling distribution once the number of samples gets large.
Your turn — statistic or parameter?
Try It Now 2.5.6 — the movie survey, again
x computed from the thirty students is a statistic. What population quantity does it estimate, and why can we not simply compute that quantity directly?
μ — the population mean movies-per-week for every student
Computing μ directly would require surveying every member of the population, which is normally impossible, too expensive, or too slow. The Law of Large Numbers is what makes the sample substitution trustworthy.
§2.5.7 — when only the intervals survive
Sometimes data arrives already summarized: you know 4 students scored somewhere between 62.5 and 68.5, but not what any one of them actually scored. You cannot compute an exact mean — but you can estimate it.
Stand in one number for every value you cannot see.
§2.5.7 — one value stands in for the whole interval
Definition 2.5.7 — Midpoint
The midpoint of an interval is
m=2lower+upper
The single value used to represent every data point falling inside that interval.
Figure: the midpoint stands in for every unknown value inside an interval.
Insight Note — you are betting the values are spread evenly
You are betting the values are spread evenly.
Using the midpoint assumes that within each interval, the data sits about as often above the middle as below it. Over several intervals those small errors tend to cancel, which is why the estimate usually lands close — and why it is still only an estimate.
§2.5.7 — the same mean, fit to what a table gives you
The idea — the frequency-weighted mean from §2.5.1, with midpoints standing in
μ=∑f∑fm
f is the frequency of an interval and m is that interval's midpoint. Read it as: multiply each midpoint by how many values sit in its interval, add those up, and divide by the total count.
Since the individual values inside a grouped table are unknown, the midpoint of each interval stands in for every value inside it — exactly the frequency-weighted trick Definition 2.5.1 introduced.
Your turn — hours spent gaming
| Hours | Teenagers |
|---|---|
| 0–3.5 | 3 |
| 3.5–7.5 | 7 |
| 7.5–11.5 | 12 |
| 11.5–15.5 | 7 |
| 15.5–19.5 | 9 |
Table 2.5.2: hours teenagers spend on video games.
μ≈10.78 hours
Midpoints 1.75, 5.5, 9.5, 13.5, 17.5. ∑fm=1.75(3)+5.5(7)+9.5(12)+13.5(7)+17.5(9)=409.75 ∑f=38μ=38409.75≈10.78
Worked example — eight grade intervals, one class mean
| Grade Interval | Students |
|---|---|
| 50–56.5 | 1 |
| 56.5–62.5 | 0 |
| 62.5–68.5 | 4 |
| 68.5–74.5 | 4 |
| 74.5–80.5 | 2 |
| 80.5–86.5 | 3 |
| 86.5–92.5 | 4 |
| 92.5–98.5 | 1 |
Table 2.5.3: professor Blount's last statistics test — grade intervals and counts.
Find the best estimate of the class mean. Every interval is a different width here — the midpoint step still works the same way on each one.
Example 2.5.5 — worked through
Weighted sum
Midpoints: 53.25, 59.5, 65.5, 71.5, 77.5, 83.5, 89.5, 95.5.
∑fm=53.25(1)+65.5(4)+71.5(4)+77.5(2)+83.5(3)+89.5(4)+95.5(1)=1460.25
Divide by the count
∑f=1+0+4+4+2+3+4+1=19 students.
μ=191460.25≈76.86
The best estimate of the class mean is 76.86 — an estimate, not an exact value, because the individual scores inside each interval were never recorded.
Key Terminology — the nine terms this section defined
Measures of the center
mean (arithmetic mean) — sum of all data values divided by the count; x for a sample, μ for a population.
median — the value splitting ordered data in half; location 2n+1.
mode — the value that occurs most frequently; the only center measure that also works on qualitative data.
bimodal — a data set with exactly two modes.
Sampling and grouped data
Law of Large Numbers — a sample mean grows very likely to approach μ as sample size increases.
statistic — a number calculated from a sample, used to estimate a population parameter.
sampling distribution — a relative frequency distribution built from a great many samples.
frequency table — grouped data displayed alongside its frequencies.
midpoint — the average of an interval's boundaries, standing in for every value inside it.
The headline result of §2.5
Three centers answer three different questions — and disagreement between them is information, not noise.
The mean balances; the median locates; the mode counts. When they land close together, any one will do. When they pull apart, the gap tells you where to look.
Every worked example in this section reduces to the same judgment call: which of these three numbers actually describes what is typical here, and which one does the extreme value or the imbalance distort?
† The Law of Large Numbers is the reason any of this arithmetic is worth trusting in the first place — a sample mean is only a useful stand-in for μ because bigger samples reliably land closer to it.
§2.5 — conclusions
The core idea
The mean adds and divides; the median orders and locates; the mode tallies and picks the winner. All three are statistics — numbers from a sample standing in for a population you cannot fully measure, and trustworthy because of the Law of Large Numbers.
The failure case
A single extreme value pulls the mean toward it without moving the median at all. Reporting only the mean of income, price, or wage data — without checking whether it agrees with the median — is how "average" becomes misleading rather than informative.
Next: §2.6 — Skewness and the Mean, Median, and Mode, where the direction the mean and median disagree becomes a picture you can read at a glance.