(a) Explain each of the following measures of central tendency (i) mean; (ii) median; (iii) mode (b) Calculate the mean, median and mode of the following se...

Answer Details

Measures of central tendency are single values that summarise a whole set of data by locating its centre. The three common measures are the mean, the median and the mode.

(a) Explanations

• Mean (arithmetic mean): the sum of all the values divided by the number of values. For values \(x_1, x_2, \dots, x_n\), \(\bar{x} = \dfrac{\sum x}{n}\). It uses every observation, so it is affected by extreme values.
• Median: the middle value when the data are arranged in order of size. With an even number of items it is the average of the two middle values. It is not distorted by extreme values.
• Mode: the value that occurs most frequently in the data. A set may have one mode, more than one mode, or none.

(b) Calculations for 21, 22, 23, 24, 25, 26, 23, 28, 29, 30, 24, 31, 34, 23 (there are \(n = 14\) numbers).

Mean: the sum of the values is 363.

Median: arrange in ascending order: 21, 22, 23, 23, 23, 24, 24, 25, 26, 28, 29, 30, 31, 34. With 14 items the median is the average of the 7th and 8th values (24 and 25).

Mode: the value 23 occurs three times, more often than any other value, so the mode is 23.