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Step 17: Statistics and probability

Describing data and distributions

Step 17 · Statistics and probability topics

Summarize a data set by its center, its spread and its shape, and pick the right summaries when the data are skewed or have outliers.

Center

The mean adds the values and divides by how many there are; it is the balance point of the data. The median is the middle value once the data are sorted, with half the data on each side.

Example: Mean and median

Find the mean and median of 3, 7, 8, 10, 22.

  1. Mean: 50 ÷ 5 = 10.
  2. Median: the middle of the five sorted values, 8.

AnswerMean 10, median 8

Example: An even count

Find the median of 4, 9, 11, 15, 20, 30.

  1. The two middle values are 11 and 15.
  2. Their average is 13.

Answer13

Spread

The range is the largest value minus the smallest. The interquartile range, IQR = Q3 − Q1, is the spread of the middle half. The standard deviation measures a typical distance from the mean.

s=∑(x−xˉ)2n−1s = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n - 1}}

Example: Standard deviation

Find the standard deviation of the sample 2, 4, 4, 6, 9.

  1. The mean is 25 ÷ 5 = 5.
  2. The squared distances from 5 are 9, 1, 1, 1 and 16, which add to 28.
  3. Divide by n − 1, which is 4, to get 7, then take the square root.

Answer√7, about 2.65

Example: Quartiles

Find the IQR of 1, 3, 4, 6, 7, 8, 10, 12.

  1. The lower half is 1, 3, 4, 6, with median Q1 = 3.5.
  2. The upper half is 7, 8, 10, 12, with median Q3 = 9.
  3. IQR = 9 − 3.5 = 5.5.

Answer5.5

Shape and outliers

A histogram shows the shape of a distribution: symmetric, skewed to the right (a long tail of large values) or skewed to the left. In a right-skewed distribution the long tail pulls the mean above the median.

30 commute times in minutes, skewed to the right: most are short, a few are long.

Example: The 1.5 × IQR rule

Q1 = 20 and Q3 = 30. Is a value of 48 an outlier?

  1. IQR = 10, so the fences are 20 − 15 and 30 + 15: 5 and 45.
  2. 48 is above the upper fence.

AnswerYes

Example: Choosing summaries

Home prices in a town are strongly skewed to the right. Which center and spread should you report?

  1. Skew and outliers drag the mean and standard deviation.
  2. The median and IQR are not moved by the long tail.

AnswerThe median and the IQR

ShapeMean and medianGood summaries
SymmetricAbout equalMean and SD
Skewed rightMean above medianMedian and IQR
Skewed leftMean below medianMedian and IQR

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Find the mean of 12, 15, 9, 20, 14.

    Show answer

    14

  2. Find the median of 5, 1, 9, 3, 7, 11.

    Show answer

    6

  3. Find the range of 18, 4, 27, 13.

    Show answer

    23

  4. Q1 = 12 and Q3 = 18. What is the upper fence for outliers?

    Show answer

    27

  5. A data set has mean 50 and median 40. Which way is it probably skewed?

    Show answer

    To the right

  6. Find the sample standard deviation of 1, 3, 5.

    Show answer

    2

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