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Statistics

Distribution Shape

Lesson

Distributions come in a few common shapes. Each one tells you where the “typical” values sit and how the unusual ones spread out.

1. Symmetric

Mirror-image shape (bell, mound). Mean and median are roughly equal. Examples: heights, IQ scores.

2. Skewed right (positive skew)

Long tail extends to the RIGHT. Peak on the left. mean>median\text{mean} > \text{median}. Examples: income, house prices.

3. Skewed left (negative skew)

Long tail to the LEFT. Peak on the right. mean<median\text{mean} < \text{median}. Examples: test scores when most students did well, age at death.

4. Uniform

All values about equally likely. Flat bars. Example: rolling a fair die many times.

Tail rule of thumb

The tail points to where the MEAN gets pulled. So in a right-skewed distribution, the mean is pulled to the right (bigger than the median). In a left-skewed distribution, mean is pulled to the left.

right skew: mode < median < mean
left skew: mean < median < mode

How to type your answer

Type the shape code: 1 symmetric, 2 skewed right, 3 skewed left, 4 uniform.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

Mound-shaped, mean≈median. Shape?\text{Mound-shaped, mean} \approx \text{median. Shape?}

Problem 2

Long tail to the right, mean > median. Shape?\text{Long tail to the right, mean > median. Shape?}

Problem 3

Long tail to the left, mean < median. Shape?\text{Long tail to the left, mean < median. Shape?}

Problem 4

All bars approximately the same height. Shape?\text{All bars approximately the same height. Shape?}

Practice

Standard problems matching the lesson.

Problem 5

Mean = 50, median = 40. Shape?\text{Mean = 50, median = 40. Shape?}

Problem 6

Mean = 25, median = 30. Shape?\text{Mean = 25, median = 30. Shape?}

Problem 7

Mean = 100, median = 100. Shape?\text{Mean = 100, median = 100. Shape?}

Problem 8

Histogram with all bars the same height.\text{Histogram with all bars the same height.}

Problem 9

Typical income distribution (a few very-high earners pull the mean up).\text{Typical income distribution (a few very-high earners pull the mean up).}

Problem 10

Test scores where MOST students did well but a few did poorly.\text{Test scores where MOST students did well but a few did poorly.}

Problem 11

Rolling a fair six-sided die thousands of times.\text{Rolling a fair six-sided die thousands of times.}

Problem 12

Heights of adult women (single population).\text{Heights of adult women (single population).}

Problem 13

A mound centered with equal-sized tails on both sides.\text{A mound centered with equal-sized tails on both sides.}

Problem 14

Histogram bin counts: 10, 8, 6, 4, 2.\text{Histogram bin counts: 10, 8, 6, 4, 2.}

Problem 15

Histogram bin counts: 2, 4, 6, 8, 10.\text{Histogram bin counts: 2, 4, 6, 8, 10.}

Problem 16

Histogram bin counts: 1, 3, 8, 12, 8, 3, 1.\text{Histogram bin counts: 1, 3, 8, 12, 8, 3, 1.}

Problem 17

House prices with a few very-expensive outliers.\text{House prices with a few very-expensive outliers.}

Problem 18

5K finish times: a few elite runners finish very fast, most clustered later.\text{5K finish times: a few elite runners finish very fast, most clustered later.}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

Mean = 70, median = 60. Shape?\text{Mean = 70, median = 60. Shape?}

Problem 20

Mean = 60, median = 70. Shape?\text{Mean = 60, median = 70. Shape?}

Problem 21

Bell-shaped, mean = median = mode = 50.\text{Bell-shaped, mean = median = mode = 50.}

Problem 22

Bar heights 5, 5, 5, 5, 5.\text{Bar heights 5, 5, 5, 5, 5.}

Problem 23

Age at death in modern populations (most live long, few die young).\text{Age at death in modern populations (most live long, few die young).}

Problem 24

Mean = 100, median = 95, mode = 90. Shape?\text{Mean = 100, median = 95, mode = 90. Shape?}

Problem 25

Mean = 100, median = 105, mode = 110. Shape?\text{Mean = 100, median = 105, mode = 110. Shape?}

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