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: Let's Learn
Read it, or press Listen and follow the words.
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 2: 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 3: 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 4: Watch an Example
One step at a time.
Watch Rosa Choose a Summary
Rosa has incomes: 21, 24, 26, 28, and 400 (in thousands).
- Step 1
The 400 is far from the rest, so it is an outlier. The mean is 99.8, above four of the five values.
Step 5: Your Turn
Practice makes it stick.
The Median
Problem 1 of 2
Values 4, 8, 9, 12, 100. What is the median?
The Skew
Problem 2 of 2
In a right-skewed distribution, is the mean above the median?
Summarize It
1 of 8
Values 2, 4, 6. What is the mean?
2 of 8
Values 1, 2, 3, 4, 90. What is the median?
3 of 8
Q1 = 10 and Q3 = 22. What is the IQR?
4 of 8
Sort each measure by whether an outlier moves it much.
Tap something to move it.
- Empty
- Empty
5 of 8
Left-skewed. Is the mean below the median?
6 of 8
Roughly symmetric with no outliers. Report the mean?
7 of 8
Values 5, 5, 5, 5. What is the standard deviation?
8 of 8
Order the four things you describe about any distribution.
- 1Center
- 2Spread
- 3Outliers
- 4Shape
Step 6: Quick Check
Show what you know.
Question 1 of 1
Values 3, 7, 8, 9, 200. What is the median?
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.