Averages, Range and Data

This lesson covers DfE content statements L2.23, L2.24 and L2.25 — calculating mean, median, mode and range; finding the mean from a grouped frequency table; and using averages and range to compare two data sets.

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Why Averages Matter

In everyday life and the workplace, you need to summarise data quickly. "On average, how long does a delivery take?" "What is the typical salary?" Averages give a single representative value for a set of data.

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The Three Averages and the Range

Measure	Definition	How to find it
Mean	The total divided by the number of values	Add all values, divide by count
Median	The middle value when ordered	Order the data, find the middle
Mode	The most common value	Find the value that appears most often
Range	The spread of the data	Highest value − Lowest value

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Mean

Mean = Sum of all values ÷ Number of values

Worked Example — Weekly Sales

Scenario: A shop records daily takings for a week: £420, £385, £510, £470, £395, £620, £540. What is the mean daily taking?

Sum = 420 + 385 + 510 + 470 + 395 + 620 + 540 = £3,340 Mean = £3,340 ÷ 7 = £477.14 (to 2 d.p.)

Worked Example — Test Scores

Scenario: A student's test scores are 72, 85, 68, 91, and 74. What mean score do they need on a sixth test to achieve an overall mean of 80?

Required total = 80 × 6 = 480 Current total = 72 + 85 + 68 + 91 + 74 = 390 Score needed = 480 − 390 = 90

Exam Tip: Questions that give you the mean and ask for a missing value are very common. Use the formula: Sum = Mean × Number of values. Then work backwards.

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Median

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

• If the number of values is odd, the median is the exact middle value.
• If the number of values is even, the median is the mean of the two middle values.

Position of the median: (n + 1) ÷ 2, where n is the number of values.

Worked Example — House Prices

Scenario: Five houses sold for: £185,000, £220,000, £195,000, £480,000, £205,000. Find the median.

Ordered: £185,000, £195,000, £205,000, £220,000, £480,000

Median = £205,000

Notice that the mean would be £257,000 — pulled up by the one expensive house. The median gives a better representation of the "typical" house price.

Worked Example — Even Number of Values

Scenario: Six employees have years of service: 2, 5, 7, 9, 12, 15. Find the median.

Position = (6 + 1) ÷ 2 = 3.5th value (between 3rd and 4th) Median = (7 + 9) ÷ 2 = 8 years

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Mode

The mode is the value that appears most frequently. There can be no mode, one mode, or more than one mode.

Worked Example — Shoe Sizes

Scenario: A shoe shop records sizes sold in one day: 6, 7, 8, 7, 9, 8, 7, 10, 8, 7, 6, 8, 7.

Size	Frequency
6	2
7	5
8	4
9	1
10	1

Mode = Size 7 (appears 5 times)

This helps the shop know which size to stock the most of.

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Range

Range = Highest value − Lowest value

The range measures the spread of the data. A larger range means more variation.

Worked Example — Temperature Variation

Scenario: Daily maximum temperatures for a week: 18°C, 21°C, 16°C, 23°C, 19°C, 24°C, 17°C.

Range = 24 − 16 = 8°C

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Choosing the Best Average

Average	Best used when...	Weakness
Mean	Data has no extreme values; you want to use all the data	Distorted by outliers
Median	Data has extreme values (outliers); you want the "middle"	Ignores extreme values
Mode	Data is categorical (e.g. favourite colour, shoe size)	May not exist; may not be central

Worked Example — Salary Comparison

Scenario: A company has 10 employees with these salaries: £22k, £24k, £25k, £25k, £26k, £27k, £28k, £30k, £35k, £120k.

Mean = £362k ÷ 10 = £36.2k (distorted by the £120k salary) Median = (£26k + £27k) ÷ 2 = £26.5k (better representation) Mode = £25k (appears twice)

A recruitment advert claiming "average salary £36.2k" would be misleading. The median of £26.5k is more representative.

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Mean from a Frequency Table

When data is presented in a frequency table, use:

Mean = Sum of (value × frequency) ÷ Sum of frequencies

Worked Example — Customer Ratings

Scenario: Customers rate a service from 1 to 5: