Statistics

Centre and spread

Mean and median

The median and mean are both measures of the centre of a set of data. They are sometimes called measures of central tendency. They provide a summary measure that attempts to describe a whole set of data with a single value that represents the middle or the centre of its distribution.

The median is the middle value when the data is in order.

The mean ( $\bar{x}$ ) (or average) is the sum of all the values in the set divided by the number of values in the data set.

$\bar{x} = \frac{\sum x_i}{n}$

e.g. Data set: 6, 4, 6, 5, 7, 0, 3
To find the median we need to arrange the values in order: 0, 3, 4, 5, 6, 6, 7
The median (middle score) is the 5
The mean of this data

$\bar{x} = \frac{0+3+4+5+6+6+7}{7} = \frac{31}{7} =$ $4.4$ (1 dp)

Since the mean includes every value in the distribution it is influenced by outliers and skewed distributions. The median is less affected by outliers and skewed data than the mean and it is usually the preferred measure of central tendency when the data is not symmetrical.

The following video from Crash Course explains more about measures of central tendencies and some examples in context.

Spread

Measures of spread describe how similar or varied the set of observed values are. Measures of spread include range, interquartile range, variance and standard deviation.

Range is the difference between the largest and smallest value in the data set.

The interquartile range (IQR) is the difference between the Upper Quartile and Lower Quartile. This describes the middle 50% of the values when they are ordered from lowest to highest. The IQR is often seen as a better measure of spread than range as it is not affected by outliers.

The variance and standard deviation are measures of the spread of the data about the mean. They summarise how close the data values are to the mean value. The smaller the variance and standard deviation, the more the mean value is indicative of the whole data set.

The variance $\sigma^2$ can be viewed as an ‘average’ distance each individual value is away from the average (mean).

The standard deviation ( $\sigma$ ) is the square root of the variance. The standard deviation of a sample can be found using the formula:

$\sigma = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}$

Your calculator or statistics software package can calculate this for you.

The following video from Crash Course explains more about measures of spread and some examples in context.

Further information

Khan academy videos explain measures of central tendency and measures of spread and how they are calculated.

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