Averages from a grouped frequency table

A grouped frequency table is used when the data is divided into intervals or groups. To summarise the data, we can calculate the averages (mean, median, mode) and the range. Each measure provides valuable insights into the data set, but calculating these from a grouped frequency table requires some additional steps compared to a simple frequency table.

1. Mean:

To calculate the mean from a grouped frequency table, we use the formula:

\[ \text{Mean} = \frac{\sum (\text{Midpoint of group} \times \text{Frequency})}{\text{Total frequency}} \]

Steps:

1. Find the midpoint of each class interval (e.g., for the interval 10–20, the midpoint is 15).
2. Multiply each midpoint by its corresponding frequency.
3. Sum all the products from step 2.
4. Sum the frequencies to find the total frequency.
5. Divide the total from step 3 by the total frequency to get the mean.

Example: For the following grouped frequency table:

\[ \text{Sum of } (m \times f) = 15 + 90 + 100 + 70 = 275 \] \[ \text{Total frequency} = 3 + 6 + 4 + 2 = 15 \] \[ \text{Mean} = \frac{275}{15} = 18.33 \]

2. Median:

The median is the middle value of the data. In a grouped frequency table, we need to find the cumulative frequency and then locate the class interval that contains the median value.

1. Find the cumulative frequency (a running total of the frequencies).
2. Use the formula to find the median position: \[ \text{Median position} = \frac{\text{Total frequency}}{2} \]
3. Locate the median class interval (the class interval that contains the median position).
4. Use the formula to calculate the median: \[ \text{Median} = L + \left( \frac{\frac{N}{2} - F}{f} \right) \times w \] where:
   • \( L \) is the lower boundary of the median class
   • \( N \) is the total frequency
   • \( F \) is the cumulative frequency before the median class
   • \( f \) is the frequency of the median class
   • \( w \) is the class width

Example: Using the same frequency table:

\[ \text{Total frequency} = 15 \quad \text{so the median position is} \quad \frac{15}{2} = 7.5 \]

From the cumulative frequency table:

From the cumulative frequency table, the median class is \(10–20\), since the cumulative frequency reaches 9 here and the median position is 7.5.

Now, use the median formula: \[ L = 10, \quad F = 3, \quad f = 6, \quad w = 10 \] \[ \text{Median} = 10 + \left( \frac{7.5 - 3}{6} \right) \times 10 = 10 + \left( \frac{4.5}{6} \right) \times 10 = 10 + 7.5 = 17.5 \]

3. Mode:

The mode is the class interval with the highest frequency. It is the most frequent value or class in the data set.

Example: From the frequency table above, the class interval with the highest frequency is \(10–20\), which has a frequency of 6. Thus, the mode is in the interval \(10–20\).

4. Range:

The range measures the spread of the data and is calculated as the difference between the largest and smallest values of the data. In a grouped frequency table, we use the boundaries of the first and last intervals.

The formula is:

\[ \text{Range} = \text{Upper boundary of the last class} - \text{Lower boundary of the first class} \]

Example: For the table above, the upper boundary of the last class (30–40) is 40, and the lower boundary of the first class (0–10) is 0. \[ \text{Range} = 40 - 0 = 40 \]

5. Summary:

• To find the mean, calculate \( \sum (m \times f) \) and divide by the total frequency.
• To find the median, use cumulative frequency to locate the median class and apply the median formula.
• To find the mode, identify the class interval with the highest frequency.
• To find the range, subtract the lower boundary of the first class from the upper boundary of the last class.