Example 1: Finding the Mean

Question: Find the mean of the following set of numbers: $4, 6, 8, 10, 12.$

Solution:

1. Add up all the values:

2. Divide the total by the number of values:

The mean is $8$.

Example 2: Finding the Median

Question: Find the median of the following set of numbers: $3, 1, 9, 7, 5.$

Solution:

3. Arrange the numbers in ascending order: $1,3,5,7,9$

4. The middle value is 5, so the median is $5$.

Note: If the data set has an even number of values, you find the mean of the two middle numbers.

Example 3: Median with Even Number of Values

Question: Find the median of the following set of numbers: $2, 4, 6, 8.$

Solution:

5. Arrange the numbers in ascending order: $2,4,6,8$

6. The middle values are $4$ and $6$. Find the mean of these two values: $\frac{4+6}{2}=5$

The median is $5$.

Example 4: Finding the Mode

Question: Find the mode of the following set of numbers: $2, 4, 4, 5, 6, 6, 6, 7$.

Solution:

• The number $6$ appears three times, more frequently than any other number. The mode is $6$.

Example: Finding the Range

Question: Find the range of the following set of numbers: $12, 7, 5, 15, 9$.

Solution:

7. The largest value is $15$.
8. The smallest value is $5$.
9. Subtract the smallest value from the largest value:

The range is $10$.

Example

Andrew Flintoff and Michael Vaughan are comparing their cricket performances. Their scores are as follows:

• Andrew Flintoff's Scores:

• Michael Vaughan's Scores:

We can calculate the Mean for both cricketers.

1. The Mean

Step 1: Add up all their scores.

• Andrew Flintoff's Total Runs:

• Michael Vaughan's Total Runs:

Step 2: Divide the total by the number of games.

• Andrew Flintoff's Games Played: $15$ games.

• Michael Vaughan's Games Played: $14$ games.

What Does This Tell Us?

• On average, Michael Vaughan has scored $46.5$ runs per game, while Andrew Flintoff has scored $36.7$ runs.
• It looks like Michael Vaughan has had the better season, based on the mean alone.

2. The Median

How to Work Out the Median:

1. Arrange the data values in ascending order.
2. Identify the middle value. If there are two middle values (even number of data points), find the mean of those two values.

Example: Comparing Cricketers Using the Median

Let's find the median for each cricketer's scores:

Andrew Flintoff's Scores:

• The median is the middle value. Since there are $15$ scores, the median is the $8th$ value:

Michael Vaughan's Scores:

• The median is the mean of the $7th$ and $8th$ values:

What does this tell us?

The median shows a different picture. Michael Vaughan's median score is $3$, which is much lower than Andrew Flintoff's median of $32$. This suggests that Michael Vaughan was much more inconsistent, scoring very low in most games but achieving a couple of exceptionally high scores.

• Good thing about the median: It is not affected by outliers, so it gives a better sense of typical performance.
• Bad thing about the median: It doesn't consider all the data points, only focusing on the middle, which can ignore important information about the rest of the data.

3. The Mode

How to Work Out the Mode:

1. Identify the most common value in the data set.
2. This value is the mode.

Example: Comparing Cricketers Using the Mode

Andrew Flintoff's Mode:

• The most frequent score is 32, which appears multiple times in his data set.

• Mode = $32$ Michael Vaughan's Mode:

• The most frequent score is $0$, which appears the most often in his data set.

• Mode = $0$ What does this tell us?

Once again, using the mode suggests that Andrew Flintoff had the better season. His most frequent score was $32$, which shows that he consistently scored decent runs. Michael Vaughan, however, had 0 as his mode, indicating that he often scored no runs in many games.

• Good thing about the mode: It's very quick and easy to find.
• Bad thing about the mode: The mode can sometimes be misleading, especially when data is skewed or has outliers. In Michael Vaughan's case, his mode is $0$ because of his many low scores, but this doesn't reflect the two very high scores of $370$ and $250$.

4. The Range

How to Work Out the Range:

1. Subtract the smallest value from the largest value.

Example: Comparing Cricketers Using the Range

Andrew Flintoff's Range:

• Largest score: $86$
• Smallest score: $10$
• Range:

Michael Vaughan's Range:

• Largest score: $370$
• Smallest score: $0$
• Range:

What does this tell us?

At first glance, someone might think that Michael Vaughan had the better season because he has the larger range of $370$, but that's not entirely correct. A larger range means that his scores varied significantly, showing inconsistency. Andrew Flintoff, with a smaller range of $76$, had a more consistent performance.

• Good thing about the range: It gives a quick idea of how spread out the data is.
• Bad thing about the range: The range is heavily affected by outliers. Michael Vaughan's very high scores of $370$ and $250$ make his range large, but they don't reflect his many low scores of $0$.

Example:

The table below shows the attendance at Preston North End football matches over a season:

Step-by-Step Guide:

1. Work out the mid-point for each group.

• The mid-point is the average of the lower and upper boundaries of each class.
• For the first group $0≤x<5,000,$ the mid-point is:

• Continue for the remaining groups.

2. Multiply the mid-point by the frequency for each group.

• This gives an estimate of the total attendance for each group of matches.

3. Calculate the estimated mean using the formula:

Step-by-Step Solution:

Totals:

• Total frequency (number of matches) = $49$
• Total of mid-point × frequency = $542,500$ Now, calculate the estimated mean: