Averages and range Revision Notes

When analysing a set of data, it is important to summarise the information using numerical measures that describe its central value and variability. The four key measures are mean, median, mode, and range. Each measure provides different insights: the mean gives an average value, the median identifies the middle point, the mode shows the most common value, and the range indicates the spread of the data.

This note focuses on calculating these measures from ungrouped data, which means data presented as individual values or frequency tables without grouping into intervals. Understanding these calculations is essential for interpreting data accurately in various contexts.

• Mean: average value of data.
• Median: middle value when data is ordered.
• Mode: most frequent value.
• Range: difference between maximum and minimum values.

The mean is found by adding all data values together and dividing by the total number of values. When data is given as a list, sum all values and divide by the count. When data is presented as a frequency table, multiply each value by its frequency, sum these products, then divide by the total frequency.

Formula: \[ \text{Mean} = \frac{\sum (x \times f)}{\sum f} \] where \(x\) is a data value and \(f\) is its frequency.

Example: Consider the data values 3, 5, 5, 7, 9. The sum is 3 + 5 + 5 + 7 + 9 = 29. There are 5 values, so mean = 29 ÷ 5 = 5.8. If given as a frequency table: Value: 3, 5, 7, 9; Frequency: 1, 2, 1, 1. Then sum of products = (3×1)+(5×2)+(7×1)+(9×1) = 3+10+7+9=29; total frequency = 5; mean = 29 ÷ 5 = 5.8.

• Sum all values or multiply values by frequencies and sum.
• Divide by total number of values or total frequency.
• Mean is sensitive to extreme values (outliers).

The median is the middle value when all data points are arranged in ascending order. If the number of data points is odd, the median is the middle value. If even, the median is the average of the two middle values.

To find the median from a frequency table, first list the data values in order, repeating each value according to its frequency, then identify the middle value(s).

Example: Data: 2, 4, 4, 6, 8. Ordered list has 5 values (odd), so median is the 3rd value: 4. For even number of values, e.g., 2, 4, 4, 6, median is average of 2nd and 3rd values: (4 + 4)/2 = 4.

• Order data values from smallest to largest.
• Identify middle value(s) depending on odd or even count.
• Median is less affected by extreme values than mean.

The mode is the value that appears most frequently in the data set. There can be more than one mode if multiple values share the highest frequency, or no mode if all values occur once.

From a frequency table, the mode corresponds to the value(s) with the highest frequency.

Example: Data: 1, 2, 2, 3, 4. Mode is 2 because it appears twice, more than any other value. If data is 1, 2, 3, 4, no value repeats, so no mode.

• Find value(s) with highest frequency.
• Mode can be unimodal, bimodal, or no mode.
• Mode is useful for categorical data or identifying common values.

The range measures the spread of the data by subtracting the smallest value from the largest value.

Formula: \[ \text{Range} = \text{Maximum value} - \text{Minimum value} \]

Example: Data: 3, 7, 8, 10, 15. Range = 15 - 3 = 12. The range gives a quick sense of variability but does not reflect how data is distributed between extremes.

• Identify maximum and minimum values.
• Subtract minimum from maximum to find range.
• Range is simple but sensitive to outliers and does not show distribution shape.

Each measure serves different purposes: the mean is useful for overall average but affected by extreme values; the median is better for skewed data or when outliers exist; the mode identifies the most common value, especially for categorical data; the range shows the spread but can be misleading if outliers are present.

In exams, carefully read whether data is ungrouped and presented as a list or frequency table. Show all working clearly, including ordering data for median, summing products for mean, and identifying frequencies for mode.

Always check calculations and state units if relevant. Use correct terminology such as 'mean is', 'median is', 'mode is', and 'range is' followed by the value.

• Use mean for balanced data without outliers.
• Use median for skewed data or when outliers exist.
• Use mode for categorical or most frequent values.
• Use range to describe spread but be cautious of outliers.
• Show clear working and label answers in exams.