Averages and Range for Ungrouped and Grouped Data

To find the median's position in an ordered list, we use:

Worked Example: Mean and Range

Five students score the following marks in a spelling test: 12, 18, 15, 12, 23. Calculate the mean and the range.

Step 1: Calculate the mean by summing the values and dividing by the number of values ($n = 5$).

• $\text{Mean} = \frac{12 + 18 + 15 + 12 + 23}{5}$
• $\text{Mean} = \frac{80}{5} = 16$ marks

Step 2: Calculate the range by subtracting the lowest score from the highest score.

• $\text{Range} = 23 - 12 = 11$ marks

Worked Example: Median and Mode

Find the median and mode of this data set: 7, 3, 9, 3, 4, 11.

Step 1: Order the data from smallest to largest.

• $3, 3, 4, 7, 9, 11$

Step 2: Identify the mode by finding the most frequent number.

• $\text{Mode} = 3$

Step 3: Find the median. Since $n = 6$ (an even number), the position is $\frac{6+1}{2} = 3.5$.

• The median is halfway between the 3rd value ($4$) and the 4th value ($7$).
• $\text{Median} = \frac{4 + 7}{2} = 5.5$

Estimating Averages for Grouped Data

When collecting heights of hundreds of plants, recording every exact millimetre takes too long, so scientists group the data into intervals. Once data is placed into a frequency table, the exact original values are hidden. Because we do not know the raw data, we can only calculate an estimated mean, and identify the modal class and median class interval.

To estimate the mean, we must assume that every data point in a class interval is equal to its exact middle, called the midpoint ($x$). We multiply each midpoint by its frequency to create a product column ($fx$), then divide the total of this column by the total frequency ($\sum f$).

The modal class is simply the interval with the highest frequency. To find the median class interval, we look for the interval containing the $\frac{n + 1}{2}$ value by keeping a running total of the frequencies, known as the cumulative frequency.

Worked Example: Estimated Mean, Modal Class, and Median Class

The table below shows the time taken ($t$, in minutes) for 30 students to travel to school. Estimate the mean time, identify the modal class, and find the median class interval.

Step 1: Add a column for the midpoint ($x$) and multiply it by the frequency ($f \times x$).

• For $0 \le t < 10$, midpoint $x = 5$. Product $fx = 6 \times 5 = 30$.
• For $10 \le t < 20$, midpoint $x = 15$. Product $fx = 14 \times 15 = 210$.
• For $20 \le t < 30$, midpoint $x = 25$. Product $fx = 10 \times 25 = 250$.

Step 2: Sum the frequency column ($\sum f$) and the product column ($\sum fx$).

• $\sum f = 6 + 14 + 10 = 30$
• $\sum fx = 30 + 210 + 250 = 490$

Step 3: Calculate the estimated mean.

• $\text{Estimated Mean} = \frac{490}{30} = 16.33$ minutes

Step 4: Identify the modal class (the interval with the highest frequency).

• $\text{Modal Class} = 10 \le t < 20$ (because it has the highest frequency of 14)

Step 5: Find the median class interval. The median position is $\frac{30+1}{2} = 15.5$.

• The first class contains values 1 to 6. The second class adds 14 more, containing values 7 to 20. The 15.5th value falls in this second group.
• $\text{Median Class Interval} = 10 \le t < 20$

Estimating the Range for Grouped Data

You can quickly approximate the spread in a grouped survey by calculating an estimated range. Because the true highest and lowest values in the data set are unknown, you must use the extremes of the table.

The standard method for GCSE is the class boundaries method. You subtract the lower bound of the very first interval in the table from the upper bound of the final interval. Because it is highly unlikely the true minimum and maximum fall exactly on these boundaries, this method is usually an overestimate of the true spread.

Worked Example: Estimated Range

Using the travel time data above, calculate the estimated range using the class boundaries method.

Step 1: Identify the upper bound of the highest class and the lower bound of the lowest class.

• Highest class is $20 \le t < 30$, so the Upper Bound is $30$.
• Lowest class is $0 \le t < 10$, so the Lower Bound is $0$.

Step 2: Subtract the lower bound from the upper bound.

• $\text{Estimated Range} = 30 - 0 = 30$ minutes