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StatisticsIntroductory Statistics

Descriptive Statistics Study Guide

Descriptive statistics summarize a data set by its center, spread and shape, and flag unusual values.

Key concepts

Center

The mean uses every value; the median is the middle value and resists outliers.

Spread

Range, interquartile range and standard deviation measure variability; the IQR resists outliers.

Shape

In a right-skewed distribution the mean is usually pulled above the median.

Standardizing

A z-score expresses a value in standard deviations from the mean, so different scales can be compared.

Definitions

Mean
Sum of the values divided by how many there are.
Median
Middle value of the sorted data (average of the two middle values if n is even).
Standard deviation
Typical distance of the values from the mean.

Formulas to know

Sample mean
xˉ=1n∑xi\bar{x} = \frac{1}{n}\sum x_i
Sample standard deviation
s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
z-score
z=x−μσz = \frac{x - \mu}{\sigma}

Worked examples

Population standard deviation

  1. Data: 2, 4, 4, 4, 5, 5, 7, 9; mean 5.
  2. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16; sum 32.
  3. Variance 32/8=432 / 8 = 4, so σ=2\sigma = 2.

Common mistakes

  • Dividing by n instead of n − 1 for a sample standard deviation.
  • Forgetting to sort before finding the median.
  • Calling the mean the typical value in strongly skewed data.

Practice questions

  1. 1.Median of 8, 3, 5, 10
    Show answer
    6.5
  2. 2.A test has mean 70 and SD 8. What is the z-score of 82?
    Show answer
    1.5

Keep practising

Find the mean, median and mode of 1, 2, 3, 4, 4.

Three measures of center for one small data set.

Find the population standard deviation of 2, 4, 4, 4, 5, 5, 7, 9.

A data set whose mean is 5 and whose variance comes out to a perfect square.

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