Averages from a frequency table

A frequency table summarises data by showing how often each value (or group of values) occurs. To summarise the data, we calculate the averages (mean, median, and mode) and the range. Each measure provides different insights into the data set.

1. Mean:

The mean is calculated using the formula:

\[ \text{Mean} = \frac{\text{Sum of (value} \times \text{frequency)}}{\text{Total frequency}} \]

Steps:

1. Multiply each value by its frequency.
2. Add these products to find the total of \((\text{value} \times \text{frequency})\).
3. Divide the total by the sum of all frequencies.

Example:

For the following frequency table:

\[ \text{Total of } (x \times f) = 6 + 20 + 12 = 38 \] \[ \text{Total frequency} = 3 + 5 + 2 = 10 \] \[ \text{Mean} = \frac{38}{10} = 3.8 \]

2. Median:

The median is the middle value when the data is arranged in ascending order. For a frequency table:

1. Calculate the cumulative frequency (a running total of the frequencies).
2. Find the position of the middle value using: \[ \text{Median position} = \frac{\text{Total frequency} + 1}{2} \]
3. Locate this position in the cumulative frequency column.

Example: For the same table:

\[ \text{Median position} = \frac{10 + 1}{2} = 5.5 \] The 5.5th value is in the cumulative frequency of 8, corresponding to the value \(4\). \[ \text{Median} = 4 \]

3. Mode:

The mode is the value with the highest frequency.

Example: From the table above, the highest frequency is \(5\), corresponding to the value \(4\): \[ \text{Mode} = 4 \]

4. Range:

The range measures the spread of the data and is calculated as the difference between the largest and smallest values. The formula is:

\[ \text{Range} = \text{Largest value} - \text{Smallest value} \]

Example: For the same table, the largest value is \(6\), and the smallest value is \(2\): \[ \text{Range} = 6 - 2 = 4 \]

5. Summary:

• To find the mean, calculate \( \sum (x \times f) \) and divide by the total frequency.
• To find the median, use cumulative frequency to locate the middle value.
• To find the mode, identify the value with the highest frequency.
• To find the range, subtract the smallest value from the largest value.