Averages and measures of spread

Maths · Statistics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

An average is one value that represents a whole set of data. The mean uses every value, so it is the usual choice, but one very large or very small value pulls it away from the typical value. The median is the middle value, so it is not affected by extreme values. The mode is the most common value and is the only average for data in categories.

The range is not an average. It measures spread: how far apart the smallest and largest values are.

When data is grouped, the exact values are not known. Use the midpoint of each class to stand for all the values in it. The result is only an estimate of the mean.

Rules to remember

Mean = sum of the values ÷ number of values.
Median = middle value when the values are in order. For n values it is the (n + 1)/2 th value; with two middle values, take the number halfway between them.
Mode = the value that occurs most often. Range = largest value − smallest value.
Frequency table: mean = sum of (value × frequency) ÷ sum of frequencies (Σfx ÷ Σf).
Grouped data: estimate of the mean = sum of (midpoint × frequency) ÷ sum of frequencies.
Modal class = the class with the highest frequency.
Sum of the values = mean × number of values.

Common mistake

In a frequency table, dividing by the number of rows instead of by the total frequency. Always divide by the sum of the frequencies, and for grouped data use the midpoints, not the upper ends of the classes.

Worked example

The masses m kg of 20 parcels are grouped: 0 < m ≤ 2 (frequency 6), 2 < m ≤ 4 (frequency 10), 4 < m ≤ 8 (frequency 4). Calculate an estimate of the mean mass.

Midpoints of the classes: 1, 3 and 6.
Sum of (midpoint × frequency) = 1 × 6 + 3 × 10 + 6 × 4 = 6 + 30 + 24 = 60.
Sum of frequencies = 6 + 10 + 4 = 20.
Estimate of the mean = 60 ÷ 20 = 3.

Answer: 3 kg

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