# Descriptive statistics

The mean, mode and median, are all ways of measuring “averages”. Depending on the distribution of the data, the values for the mean, mode and median can differ slightly or a lot. Therefore, the mean, mode and median are all useful for understanding your data set.

Example data set: 6, 3, 6, 13, 7, 7 in a table:

Mean  the average value,

Mode  the value that occurs most often (highest frequency) e.g. The example data set has 2 modes: 6 and 7

Median the middle value when the data set is ordered low to high. Even number of values: the median is the average of the two middle values. Find for larger values as n + ½ .

e.g. data set from low to high: 3, 6, 6, 7, 7, 13

Range largest x-value − smallest x-value

e.g. range=13−3=10

Variance

Standard deviation

Grouped data data presented as an interval, e.g.10 < x ≤ 20 where:

• lower boundary = 10
• upper boundary = 20
• interval width = 20 − 10 = 10
• mid-interval value (midpoint) = (20 + 10)/2 = 15

Use the midpoint as the x-value in all calculations with grouped data.

Adding a constant to all the values in a data set or multiplying the entire data set by a constant influences the mean and standard deviation values in the following way:

Table 7.1: Adding or multiplying by a constant

Q1   the value for x so that 25% of all the data values are ≤ to it first quartile   = 25th percentile

Q2   median   = 50th percentile

Q3   third quartile   = 75th percentile

Q3 − Q1   interquartile range (IQR)   = middle 50 percent

Example: Snow depth is measured in centimetres: 30, 75, 125, 55, 60, 75, 65, 65, 45, 120, 70, 110. Find the range, the median, the lower quartile, the upper quartile and the interquartile range.

First always rearrange data into ascending order: 30, 45, 55, 60, 65, 65, 70, 75, 75, 110, 120, 125

1. The range

125−30=95cm

2. The median: there are 12 values so the median is between the 6th and 7th value.

(65 + 70) / 2   = 67.5 cm

3. The lower quartile: there are 12 values so the lower quartile is between the 3rd and 4th value.

(55 + 60) / 2   = 57.5 cm

4. The upper quartile: there are 12 values so the lower quartile is between the 9th and 10th value.

(75 + 110) / 2   = 92.5 cm

5. The IQR

92.5 − 57.5 = 35cm