Averages and Spread
☰ Contents
An average is a single number standing in for many, and the whole topic is about what that substitution costs. Two data sets can share a mean and have nothing else in common — which is why every serious statement about data quotes a center and a spread, and why "the average" on its own is the most quietly misleading phrase in statistics.
Which average should you use?
Use the mean when the values are spread fairly evenly, the median when the data has outliers or is lopsided, and the mode when the data is in categories, where there is no arithmetic to do. Measures of Central Tendency makes that choice a deliberate one. The mean adds the values and divides by how many there are. For 4, 5, 5, 6 and 30, the mean is (4 + 5 + 5 + 6 + 30) ÷ 5 = 10, but the median, the middle value, is 5, and the mode, the most common value, is also 5. The mean moved because the one large value, 30, pulls the total up, while the median depends only on the order. With an even number of values, the median is halfway between the middle two: for 3, 5, 7 and 9 it is (5 + 7) ÷ 2 = 6. A mean must sit between the smallest and largest values.
the median is the middle value once the data is in order, so it depends on rank and not on size — moving x₅ past the others never moves it
Drag the last value away from the rest to make it an outlier
The Mean shares the total out equally, and Median and Mode gives the other two.
Salaries of 22, 24, 25, 26 and 180 (thousands): the mean is 55.4, the median is 25. Nobody in that office earns anything like 55.4, and the median is the honest summary.
Averages from a table
When the same values repeat, listing them all is wasteful. Averages from a Frequency Table shows the shortcut: multiply each value by its frequency, total those products, and divide by the total frequency. The median comes from a running total of frequencies, which is where the halfway item falls.
Grouped data is harder, because the individual values are gone. Mean of Grouped Data assumes every value sits at its class midpoint, which makes the answer an estimate — an honest one, and it should be labeled as such. The Modal Class makes the parallel point on the other side: with grouped data you can name the busiest class, but not a single modal value.
Now you
Which average is barely moved by one very small value?
Which average is barely moved by one very large value?
Why is an average not enough?
An average gives the center of the data but says nothing about how spread out the values are, and the spread is often what matters. Measures of Spread shows this with two data sets. Take two players' scores over three games: one scored 9, 10 and 11, and the other scored 1, 10 and 19. Both means are 10, since 9 + 10 + 11 = 30 and 1 + 10 + 19 = 30. The simplest measure of spread is the range, the largest value minus the smallest. The first player's range is 11 − 9 = 2 and the second player's is 19 − 1 = 18, so the first player is far more consistent. A coach choosing a reliable player needs the spread. The range has a weakness: one unusual value can make it huge, which is why the interquartile range is often used instead. Sort the data before finding a range, since an unsorted list hides its largest and smallest values.
20/40 are below this, so it is the median
Read across at a quarter of the total, and drop to the value.
Range and IQR introduces the fix for the range. The interquartile range is the upper quartile minus the lower quartile, so it describes the middle half of the data and one extreme value cannot touch it.
Percentiles generalizes the idea: the 90th percentile is the value that 90% of the data falls below. Quartiles are just the 25th, 50th and 75th percentiles under an older name, and percentiles are how growth charts, exam grades and response-time targets are all actually reported.
Cumulative frequency and quartiles
Cumulative Frequency is a running total of counts, and plotting it against the upper boundary of each class gives the S-shaped curve that answers "how many were below this value" for any value at all.
Quartiles from the Curve reads the quartiles off it directly: go up the vertical axis to a quarter of the total, across to the curve, and down to the value. The same move at half and three-quarters gives the median and the upper quartile.
Now you
The lowest value is 4 and the highest is 29. What is the range?
The readings are 14, 18 and 23, with one outlier at 48. What is the range?
What does a box plot show?
A box plot shows five numbers: the minimum, the lower quartile, the median, the upper quartile and the maximum. Box and Whisker draws the box from the lower quartile to the upper quartile, marks the median inside it, and runs whiskers out to the smallest and largest values. For the data 3, 5, 6, 8, 10, 11 and 14, the median is 8. The lower quartile is the middle of the lower half, 3, 5 and 6, so it is 5, and the upper quartile is the middle of 10, 11 and 14, so it is 11. The box runs from 5 to 11, and its length, 11 − 5 = 6, is the interquartile range. The box holds the middle half of the data. An outlier, more than one and a half box lengths beyond the box, is drawn as a separate cross. Read the five numbers from left to right: they must be in order.
median − q1 = q3 − median at every spread, so symmetry is evidence and never proof
Keep both halves of the box equal and make the four bands the same width.
the box spans Q₁ to Q₃, IQR = 15.5, and holds ten of the twenty values however narrow it gets: a narrow box is density, not fewer points
Pack the middle ten together, then push the top value past the fence
Its value is comparison. Two box plots on the same axis show at a glance which group is higher, which is more consistent, and whether either is lopsided — three questions that would take a paragraph in words.
Judging Normality from a Box Plot turns that reading into a test: in a symmetric distribution the median sits centrally in the box and the whiskers are of similar length, so a median pushed toward one end, or one long whisker, is evidence of skew. That matters later, because several statistical procedures assume symmetry before they are allowed to run.
Now you
The lower quartile is 12 and the upper quartile is 24. How wide is the box?
100 values are drawn as a box plot. How many of them sit inside the box?
How is standard deviation calculated, and why that way?
To find the standard deviation, find how far each value is from the mean, square those distances, find the mean of the squares, and take the square root. Standard Deviation builds it step by step. For 2, 4, 4, 4, 5, 5, 7 and 9, the mean is 40 ÷ 8 = 5. The distances from 5 are −3, −1, −1, −1, 0, 0, 2 and 4, and their squares add to 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32. The mean of the squares is 32 ÷ 8 = 4, and the standard deviation is its square root, 2. Squaring is needed because the distances above and below the mean always cancel to 0, and a square is never negative. The square root returns the answer to the units of the data. If every value is the same, the standard deviation is 0. The answer should never be more than the range.
Σ(x − x̄) = 0 for every data set, so the signs come off before the average measures anything
Try to make the two sides of the balance different lengths.
(x − μ)² = 31.4: twice the distance is twice the bar and four times the square, so one far value outweighs several near ones
Drag the last value until its square reaches 100
A gentler cousin comes first. Mean Absolute Deviation averages the distances without squaring, using absolute values to stop the positives and negatives canceling. It is easier to explain and genuinely useful — and it makes the natural question sharp: if this works, why square anything?
Standard Deviation: Ungrouped works the practical formula — the mean of the squares minus the square of the mean — which gets the same answer in one pass through the data instead of two. Standard Deviation: Grouped extends it to frequency tables, weighting each squared value by its frequency and, for grouped data, using class midpoints with the same estimate caveat as the grouped mean.
Now you
What is the standard deviation of 2, 2, 10, 10?
The mean is 6 and a value is 4. What is the squared distance from the mean?
How do you compare two data sets honestly?
To compare two data sets fairly, quote one center and one spread from each, use the same pair for both, and say what the numbers mean in the context of the data. Comparing Data Sets practices exactly that structure. The median goes with the interquartile range, and the mean goes with the standard deviation. Suppose two classes take the same test. Class A has a median of 62 and an interquartile range of 10; Class B has a median of 55 and an interquartile range of 25. A good comparison says that Class A scored higher on average, by 62 − 55 = 7 marks, and that its scores were more consistent, since its middle half spans 10 marks instead of 25. When a data set has outliers, use the median pair, which outliers barely move. A comparison that lists four numbers and draws no conclusion answers nothing, so reread yours and make sure every number says something about the classes.
A wins both rows and A% > B% overall, because the two mixes are close
Make A better in both rows and worse overall.
Mixing the pairs — a median and a standard deviation — quotes a robust center against a non-robust spread, so an outlier affects half the description.
And then there is the trap that survives every correct calculation. Simpson's Paradox shows two groups where treatment A beats treatment B in the mild cases and in the severe cases, yet loses overall — because the groups were different sizes and the severe cases went disproportionately to one treatment. Nothing in the arithmetic is wrong. The lesson is that an average over a mixed population is not a property of anyone in it.
Now you
Two sets both average 25. One has a much smaller spread. Which is more consistent?
Two sets both average 30. One has a much smaller spread. Which is more consistent?
The mistakes worth naming
- Reporting a grouped mean as exact. It is an estimate built on the midpoint assumption. Say so.
- Forgetting to sort before finding a median. The middle of an unsorted list is not the middle of the data.
- Pairing a median with a standard deviation. Median with IQR, mean with standard deviation — keep the pair robust or not, but not half of each.
- Plotting cumulative frequency at class midpoints. The running total is only true at the upper boundary.
- Comparing two groups on center alone. Equal means with unequal spreads is the case where the spread is the whole story.
- Deleting an outlier because it is inconvenient. An outlier is a data point until there is a recorded reason to doubt it.
Where this leads next
These summaries are the vocabulary the rest of statistics is written in. The charts and displays guide covers the pictures they are read from, including the cumulative frequency curve and the histogram. Once each subject carries two measurements instead of one, the question becomes whether they move together — the scatter, correlation and regression guide. And the mean and standard deviation of a sample, used to say something about a population you did not measure, is the whole subject of the sampling and inference guide.
Your turn
Three to try — tap what you get.
The mean of 4, 7 and 10
The median of 3, 9 and 5
The range of 12, 4 and 9
Questions and answers
- When should you use the median instead of the mean?
- Use the median whenever the data has extreme values or a long tail, because the mean is dragged toward outliers and the median is not. House prices and incomes are the standard examples: a few very large values pull the mean above what most people experience, while the median names the middle household. Use the mean when the distribution is roughly symmetric, or when you need a total, since the mean multiplied by the count gives the sum back.
- What does standard deviation actually measure?
- It measures the typical distance of a value from the mean. You find each deviation from the mean, square them so positives and negatives cannot cancel, average the squares, then square-root to return to the original units. A standard deviation of 4 cm means values sit roughly 4 cm from the mean on average. It is in the same units as the data, which is why it is quoted rather than the variance.
- How do you find the interquartile range?
- Sort the data, find the median, then find the median of the lower half and the median of the upper half. Those are the lower and upper quartiles, and the interquartile range is the upper minus the lower. It describes the spread of the middle half of the data and ignores the extremes entirely, which is exactly why it is quoted alongside the median rather than the mean.
- Why can't you find an exact mean from a grouped frequency table?
- Because grouping throws the original values away. A class recorded as 20 to 30 might contain values clustered at 21 or at 29, and the table no longer knows which. The standard method assumes each value sits at the midpoint of its class, which is reasonable on average but not exact, so a grouped mean is always reported as an estimate. The same limitation makes the modal class a class rather than a single mode.
- What is Simpson's paradox?
- It is when a trend that holds in every subgroup reverses once the subgroups are pooled. A treatment can have a higher success rate than another in both mild and severe cases, yet a lower rate overall, because the groups have very different sizes and severities. The paradox is a warning that an average over a mixed population is not a property of any individual in it, and that combining data can invent a conclusion neither part supports.
- How do you compare two sets of data properly?
- Compare one measure of center and one measure of spread, and say what each means in context. Two classes with the same mean test score can be completely different if one has a standard deviation of 3 and the other 15, since the second contains both the strongest and the weakest students. A comparison that quotes only an average has answered half the question, and it is the half that hides the interesting difference.