Describe every distribution the same way: shape, center, spread, and any outliers. Skew drags the mean toward the tail; the median stays put. The median and IQR are resistant; the mean and standard deviation are not. Skewed or with outliers: median and IQR. Roughly symmetric: mean and SD.
Step 1: Picture It
Start with something you already know.
Coins on a long plank
- Five identical coins sit on a long, featherweight plank, at 2, 3, 4, 5 and 6 inches.
- The plank balances on a fingertip placed at 4 inches.
- The coin at 6 inches slides all the way out to 16 inches.
- Now the plank only balances with the fingertip at 6 inches.
- The middle coin, third from the left, is still at 4 inches.
The mean is the balance point: one far-off value tips it toward that side, toward the tail.
The median is the middle coin: it cares about order, not distance, so it stays put. That is what resistant means.
Where the picture falls short: a value that crosses the middle can move the median one place along the list, but sliding a value far out never carries the median with it.
The same five coins, with the last one moved further each time.
| Coin positions (inches) | Mean and median |
|---|---|
| 2, 3, 4, 5, 6 | mean 4, median 4 |
| 2, 3, 4, 5, 16 | mean 6, median 4 |
| 2, 3, 4, 5, 46 | mean 12, median 4 |
One coin moved the mean from 4 to 12. The median never left 4.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Five friends get a weekly allowance of $5, $6, $7, $8 and $49. Their mean allowance is $15. How many of the five friends get more than $15?
friends
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Shape
Symmetric, skewed right with a long tail of high values, skewed left, or with two peaks. Skew is named for the tail. Incomes, house prices and waiting times are usually right-skewed.
Resistance, measured
Drop the 400 and add 30: mean 25.8, median 26. The mean moved by 74; the median by nothing. A single value can own the mean; nothing owns the median.
Spread
The IQR for 21, 24, 26, 28, 400 is Q3 − Q1 = 28 − 24 = 4, describing the middle half. The SD is about 168, describing almost nothing but the outlier. Match the spread measure to the center measure.
Outliers
The 1.5 × IQR rule: fences at 24 − 6 = 18 and 28 + 6 = 34. 400 is far outside. Investigate it: a typing error, or a real high earner. Either way, report it separately.
The one-sentence description
"The distribution of incomes is strongly right-skewed with a median of 26 thousand, an IQR of 4 thousand, and one high outlier at 400 thousand." Shape, center, spread, outlier, in that order, in context.
Decide the shape first
The shape is what makes the choice of summary. Quoting a mean for badly skewed data is the commonest error in a description, and deciding the shape first prevents it.
Step 3: Try It Yourself
Tap and try it out.
Reaction times in ms: 210, 220, 225, 230, 235, 240, 250, 480. Describe the distribution and choose the summary
- 1seven values from 210 to 250, one at 480: skewed right by an outliershape first
Step 0 of 5
Low
Mid-low
Mid-high
High
There are 16 in all.
Low has the most. It has 7 more than High.
Step 4: In Real Life
House prices in a town
A town’s house prices are skewed right by a few mansions. The median, not the mean, is the summary the realtor quotes. Shape decides which center to report.
Step 5: Watch an Example
One step at a time.
Watch Rosa Describe Bus Waits
Rosa timed how long 8 riders waited for a bus, in minutes: 2, 3, 4, 4, 5, 6, 7 and 33. She wants one sentence that describes the waits.
- Step 1
Shape comes first. Seven waits sit between 2 and 7 minutes and one sits far off at 33, so the long tail runs toward the high values: the waits are skewed right.
Watch Dev Describe Quiz Scores
Dev has 8 quiz scores out of 100, in order: 41, 78, 82, 85, 86, 88, 90 and 94. He wants to describe them and pick the right summary.
- Step 1
The scores are already in order, so the shape can be read straight off: seven scores sit between 78 and 94, and one sits far below them at 41.
Step 6: Your Turn
Practice makes it stick.
Rents on One Street
Problem 1 of 17
Six apartments on a street rent for $900, $950, $1,000, $1,050, $1,100 and $2,800 a month. A listing site reports the mean rent. How many dollars higher is the mean than the median?
dollars
Pizza Delivery
Problem 2 of 17
A pizza shop timed 8 deliveries, in minutes: 18, 20, 21, 22, 23, 25, 26 and 52. Using the 1.5 × IQR rule, what is the upper fence, in minutes?
minutes
Summarize It
Decide the shape first when it matters. Each one is a single move from the lesson.
Problem 3 of 17
Values 2, 4, 5, 7 and 42. What is the mean?
Problem 4 of 17
Eight values in order: 10, 12, 13, 15, 16, 18, 19 and 40. What is the IQR?
Problem 5 of 17
A data set has Q1 = 40 and Q3 = 50. By the 1.5 × IQR rule, where is the lower fence?
Problem 6 of 17
Commute times in a city are skewed right, with a median of 30 minutes. Which of these could be the mean?
Problem 7 of 17
The heights of 30 sunflowers are roughly symmetric, with no outliers. Which pair of measures describes them best?
Whose Step?
Each page has one line that slipped. Find it, then put it right.
Problem 8 of 17
Five houses sold for 180, 200, 210, 230 and 900 thousand dollars. Leo summarized the prices. One line is a slip. Which?
Leo’s working: five house prices, in thousands of dollars
- 1900 is far above the rest: skewed right with an outlier
- 2median = 210
- 3mean = 1720/5 = 344
- 4skewed with an outlier, so report the mean, 344
- Answer"Skewed right, center 344 thousand dollars, one high outlier at 900"
Problem 9 of 17
What center should Leo report for the five house prices, in thousands of dollars?
thousand dollars
Problem 10 of 17
A data set has Q1 = 12 and Q3 = 20. Maya checked whether a value of 31 is an outlier. One line is a slip. Which?
Maya’s working: is 31 an outlier?
- 1Q1 = 12, Q3 = 20
- 2IQR = 20 − 12 = 8
- 3upper fence = 20 + 8 = 28
- 431 is above 28
- Answer31 is an outlier
Problem 11 of 17
With Q1 = 12 and Q3 = 20, where is the upper fence by the 1.5 × IQR rule?
Problem 12 of 17
At a children’s party, most of the 40 guests are 6 to 8 years old, and a few parents are 35 to 45. Ana described the shape of the ages. One line is a slip. Which?
Ana’s working: the shape of 40 guests’ ages
- 1most ages bunch at 6 to 8
- 2a few ages stretch out to 45
- 3the bunch is on the left, so the shape is skewed left
- Answerskewed, so report the median and IQR
Problem 13 of 17
What is the shape of the 40 guests’ ages at the party?
Pick the Move
Which move comes first, or which statement does the job? No full working is needed.
Problem 14 of 17
A new data set arrives and you must choose between reporting the mean and the median. What should you decide first?
Problem 15 of 17
The commute times of 30 workers are skewed right, with one very long commute. Which statement is a complete description?
Same Summary?
Judge each one by its shape, without working anything out in full.
Problem 16 of 17
The incomes 21, 24, 26, 28 and 400 thousand are summarized by the median and IQR. Select every data set that should be summarized the same way.
Problem 17 of 17
List A is 3, 5, 7, 9, 11. List B is 3, 5, 7, 9, 61. Without working any of them out, which measure is the same for both lists?
Step 7: Quick Check
Show what you know.
Question 1 of 1
Values 3, 7, 8, 9, 200. What is the median?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Sam wants five values with a median of 10 and a mean of 20. Four of them are 8, 9, 10 and 11. What must the fifth value be?
Stretch 2 of 2
Ravi’s eight values are 7, 8, 9, 10, 11, 12, 13 and 14. He replaces the 14 with a larger whole number. What is the smallest whole number he could use that would be a high outlier by the 1.5 × IQR rule?
What You Learned
- Describe a distribution by shape, center, spread, and outliers, in that order.
- Skew drags the mean towards the tail; the median stays put.
- Report median and IQR when skewed, mean and standard deviation when symmetric.
- Decide the shape before choosing the summary.
For the grown-up
The misconception this lesson targets is that "the average" is always the mean and always the number to report. On skewed data or data with an outlier, the mean is dragged toward the tail and can sit above nearly every value, as the $15 allowance does in the opener. The habit that cures it is deciding the shape first, out loud, before any center is chosen. A close cousin is naming skew for where the values bunch rather than for the tail: when a student says "skewed left" about incomes, ask which way the long tail points.