4 Statistics
4.1 Measures of central tendency
For a data set \(x_1, x_2, \dots, x_n\) the mean is
\[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i. \]
The median is the middle value once the data are ordered, and the mode is the most frequently occurring value.
The mean uses every value but is dragged towards outliers. The median is resistant to outliers, which is why income and house-price data are usually reported as medians.
4.2 Measures of dispersion
The standard deviation of a population is
\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}. \]
The interquartile range is \(\mathrm{IQR} = Q_3 - Q_1\). A value is commonly treated as an outlier if it lies below \(Q_1 - 1.5\,\mathrm{IQR}\) or above \(Q_3 + 1.5\,\mathrm{IQR}\).
Example 4.1 (Effect of a linear transformation) A data set has mean \(50\) and standard deviation \(8\). Each value is multiplied by \(3\) and then decreased by \(10\). Find the new mean and standard deviation.
Solution. If \(y = 3x - 10\) then
\[ \bar{y} = 3\bar{x} - 10 = 3(50) - 10 = 140, \]
\[ \sigma_y = |3|\,\sigma_x = 3(8) = 24. \]
Adding a constant shifts the data but does not spread it out, so it leaves the standard deviation unchanged.
4.3 Standard scores
The standard score of a value \(x\) is
\[ z = \frac{x - \bar{x}}{\sigma}, \]
measuring how many standard deviations \(x\) lies from the mean. Standard scores allow results from different tests to be compared directly.
4.4 Exercises
- Find the mean and median of \(4, 7, 7, 9, 13, 20\).
- A student scores \(72\) on a test with mean \(60\) and standard deviation \(8\). Find the standard score.
- Every value in a data set is increased by \(5\). State the effect on the mean, the median and the standard deviation.
Answers
- Mean \(10\), median \(8\).
- \(z = 1.5\).
- Mean and median both increase by \(5\); the standard deviation is unchanged.