Most tested M6.3

Mean Median Mode and Range

Calculating and interpreting fundamental statistical measures for both raw (ungrouped) and categorised (grouped) data is essential for summarising datasets and making justified comparisons between them, a common task in scientific and engineering analysis.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • For ungrouped data, you can calculate exact values for mean, median, mode, and range.
  • For grouped data, where you only have frequency counts for class intervals, you can only calculate *estimates* of the mean, median, and range. This is because you don't know the exact values within each group.
  • Measures of central tendency (mean, median, mode) describe the 'typical' or 'average' value in a dataset.
  • Measures of spread (range, interquartile range) describe how consistent or varied the data is.
  • When comparing two datasets, a robust answer requires commenting on both a measure of average (like the median) and a measure of spread (like the interquartile range).
  • Extreme values, or outliers, have a significant impact on the mean and range, but very little effect on the median and interquartile range.

Formulae

Mean = (Sum of values) / (Number of values)

To calculate the mean for a set of ungrouped data.

Median position = (n + 1) / 2

To find the position of the median in an ordered list of 'n' values.

Estimated Mean = Sum(f × x) / Sum(f)

To estimate the mean from a grouped frequency table, where 'f' is the frequency of a class and 'x' is its midpoint.

Definitions

Mean
The sum of all data values divided by the number of data values. It is sensitive to outliers.
Median
The middle value when the data is arranged in ascending order. If there is an even number of values, it is the average of the two middle values.
Mode
The most frequently occurring value in a dataset. A dataset can have one, more than one, or no mode.
Range
The difference between the highest and lowest values in a dataset. It is a simple measure of spread.
Interquartile Range (IQR)
The difference between the upper quartile (75th percentile) and the lower quartile (25th percentile). It describes the spread of the middle 50% of the data.

Worked example

The times, in seconds, for a group of 8 students to solve a puzzle are: 45, 51, 42, 68, 49, 42, 55, 47. Calculate the mean, median, and range of the times.

  1. 1

    First, order the data to find the median and range:

    42, 42, 45, 47, 49, 51, 55, 68.

  2. 2

    Calculate the range:

    Highest value - Lowest value = 68 - 42 = 26 seconds.

  3. 3

    Find the median position:

    There are 8 data points, so the median is the average of the (8+1)/2 = 4.5th value, i.e., the 4th and 5th values.

    These are 47 and 49.

    Median = (47 + 49) / 2 = 96 / 2 = 48 seconds
  4. 4

    Calculate the sum for the mean:

    42 + 42 + 45 + 47 + 49 + 51 + 55 + 68.

    A quick way to sum is to group them:

    (42+68) + (45+55) + (42+47+49+51) = 110 + 100 + (89+100) = 210 + 189 = 399.

  5. 5

    Calculate the mean:

    Sum / Count = 399 / 8

    To do this without a calculator, note that 400 / 8 = 50.

    Since we have 399, it's 1 less than 400.

    So the mean is 50 - 1/8 = 49.875 seconds.

Answer: Mean = 49.875s, Median = 48s, Range = 26s

Common mistakes

  • ×Simple arithmetic errors when summing a list of numbers for the mean are very common under time pressure. Double-check your addition.
  • ×Forgetting to order the data before finding the median or range. This is a critical first step.
  • ×When finding the median of an even number of data points, students often pick one of the two middle numbers instead of calculating their average.
  • ×Calculating the range from the frequencies in a table, rather than from the actual data values (i.e., the highest and lowest possible values).

No-calculator tips

  • ✓To find the mean of numbers that are close together (e.g., 995, 1002, 998), use an 'assumed mean'. Guess a round number (e.g., 1000), find the mean of the differences (-5, +2, -2), and add this to your guess. Mean of diffs = (-5)/3. So Mean = 1000 - 5/3.
  • ✓When summing numbers, look for pairs that add up to round numbers (e.g., 47 + 53 = 100) to simplify the calculation.
  • ✓For grouped data, the midpoint of a class interval like '10 ≤ x < 20' is just the average of the boundaries: (10 + 20) / 2 = 15. Calculate this quickly for each class.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 1 topics