Statistics

Averages and Range

Year 7 · Year 8 · Year 9

✓By the end of this lesson students will be able to calculate the mean, median, mode, and range for a set of discrete data.
✓By the end of this lesson students will be able to identify the most appropriate average to use in different contexts.
✓By the end of this lesson students will be able to calculate the mean, median, mode, and range from a frequency table.
✓By the end of this lesson students will be able to understand the advantages and disadvantages of each average.

Key concepts

Mean

The mean is the most commonly used average. It is calculated by adding up all the values in a data set and then dividing by the total number of values. It is sometimes referred to as the 'arithmetic mean'.

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

Median

The median is the middle value in a data set when the values are arranged in order of size (either ascending or descending). If there is an even number of values, the median is the mean of the two middle values.

Position of Median = (Number of values + 1) / 2

Mode

The mode is the value that appears most frequently in a data set. A data set can have one mode (unimodal), more than one mode (multimodal), or no mode if all values appear with the same frequency.

Range

The range is a measure of spread or dispersion of a data set. It is the difference between the highest value and the lowest value in the data set. A larger range indicates greater variability in the data.

Range = Highest value - Lowest value

Frequency Table

A frequency table is a way of organising data by listing each distinct value and the number of times it appears (its frequency). This makes it easier to calculate averages and the range, especially for larger data sets.

Calculating Mean from a Frequency Table

To calculate the mean from a frequency table, you multiply each value by its frequency, sum these products, and then divide by the total frequency (which is the sum of all frequencies).

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

Calculating Median from a Frequency Table

To find the median from a frequency table, first find the total number of data points (sum of frequencies). Then, determine the position of the median using (Total frequency + 1) / 2. Use cumulative frequency to locate which value corresponds to this position.

Position of Median = (Total frequency + 1) / 2

Calculating Mode from a Frequency Table

The mode from a frequency table is simply the value that has the highest frequency.

Key facts to remember

1The mean is the sum of all values divided by the number of values.
2The median is the middle value when data is ordered.
3The mode is the most frequent value.
4The range is the difference between the highest and lowest values.
5For an even number of data points, the median is the mean of the two middle values.
6The mean can be affected by extreme values (outliers), while the median is less affected.
7The mode is useful for categorical data or when identifying the most popular item.
8The range is a simple measure of spread but can be heavily influenced by outliers.

Worked examples

Example 1

Find the mean, median, mode, and range for the following set of data: 5, 2, 8, 3, 5, 7.

IOrder the data: 2, 3, 5, 5, 7, 8

IIMean: (2 + 3 + 5 + 5 + 7 + 8) / 6 = 30 / 6 = 5

IIIMedian: There are 6 values. The median is the mean of the 3rd and 4th values. (5 + 5) / 2 = 10 / 2 = 5

IVMode: The value 5 appears twice, which is more than any other value. So, the mode is 5.

VRange: Highest value - Lowest value = 8 - 2 = 6

Answer

Mean = 5, Median = 5, Mode = 5, Range = 6

Example 2

A group of students scored the following marks in a test: 12, 15, 10, 18, 15, 13, 15, 11. Find the mean, median, mode, and range.

IOrder the data: 10, 11, 12, 13, 15, 15, 15, 18

IIMean: (10 + 11 + 12 + 13 + 15 + 15 + 15 + 18) / 8 = 109 / 8 = 13.625

IIIMedian: There are 8 values. The median is the mean of the 4th and 5th values. (13 + 15) / 2 = 28 / 2 = 14

IVMode: The value 15 appears three times, which is more than any other value. So, the mode is 15.

VRange: Highest value - Lowest value = 18 - 10 = 8

Answer

Mean = 13.625, Median = 14, Mode = 15, Range = 8

The mean does not have to be one of the original data values.

Example 3

The number of goals scored by a football team in their last 20 matches is shown in the frequency table below. Calculate the mean, median, mode, and range for the number of goals scored. | Goals scored (x) | Frequency (f) | |------------------|---------------| | 0 | 3 | | 1 | 5 | | 2 | 8 | | 3 | 3 | | 4 | 1 |

IAdd columns for (x * f) and Cumulative Frequency:

III|-----------|----------|-------|-----------------|

IV| 0 | 3 | 0 | 3 |

V| 1 | 5 | 5 | 8 |

VI| 2 | 8 | 16 | 16 |

VII| 3 | 3 | 9 | 19 |

VIII| 4 | 1 | 4 | 20 |

9| **Total** | **20** | **34**| |

10Mean: Sum of (x * f) / Sum of f = 34 / 20 = 1.7

11Median: Total frequency (N) = 20. Position of median = (N + 1) / 2 = (20 + 1) / 2 = 10.5th value. Using cumulative frequency: The 10.5th value falls within the '2 goals' category (as cumulative frequency 8 is up to 1 goal, and cumulative frequency 16 is up to 2 goals). So, the median is 2.

12Mode: The highest frequency is 8, which corresponds to 2 goals. So, the mode is 2.

13Range: Highest value - Lowest value = 4 - 0 = 4

Answer

Mean = 1.7, Median = 2, Mode = 2, Range = 4

When calculating the median from a frequency table, the median value itself will be one of the 'x' values, not the position.

Common mistakes

✗Not ordering the data before finding the median.
✗Confusing the definitions of mean, median, and mode.
✗Forgetting to divide by the total number of values when calculating the mean.
✗Forgetting to multiply the value by its frequency when calculating the mean from a frequency table.
✗Incorrectly identifying the median from an even number of data points (e.g., picking one of the two middle values instead of their mean).
✗Calculating the range by adding the highest and lowest values instead of subtracting.

Exam tips

★Always write down the data in ascending order before attempting to find the median.
★Show all steps in your calculations, especially for the mean, as this can earn method marks even if the final answer is incorrect.
★For frequency tables, consider adding an 'x * f' column and a 'cumulative frequency' column to help with calculations.
★Read the question carefully to determine which average is being asked for, or if you need to choose the most appropriate one.
★Double-check your arithmetic, especially when summing many numbers or multiplying in frequency tables.

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