Methods to explain your results

Example

As part of a school project, Kieran is asked to write down the number of tracks on each of his CDs. His results are as follows:

10, 14, 10, 12, 10, 11, 12, 10, 11, 9 and 12.

This gives Kieran the information he needs, but the data is not very easy to read or remember. What can he do to improve on this? One way is to put his results into a table:

This is better - but it would still be difficult to compare his results with those of his classmates.

Kieran therefore decides to describe his results by giving the average number of tracks, and some indication of the spread of his results.

The mean

The mean is the most common measure of average. If you ask someone to find the average, this is the method they are likely to use.

Kieran's results were:

10  14  10  12  10  11  12  10  11  9  12

The mean for this example is:

Kieran only had 11 results, so calculating the mean in this way was not too time consuming or complicated.

Finding the mean from a frequency table

If Kieran had 111 CDs, it is likely that he would have made a mistake while typing the results into his calculator. In cases like this, finding the mean from a frequency table is more efficient.

Work through an example in the activity below to practise reading information from a frequency table.

The median

The median is the middle number. To calculate the median of any set of numbers, you need to write the numbers in order.

To find the median number:

• Put all the numbers in numerical order.
• If there is an odd number of results, the median is the middle number.
• If there is an even number of results, the median will be the mean of the two central numbers.

Finding the median with an odd number of results

Using the same example, find the median number of tracks on Kieran's CDs.

Kieran's results were:

10  14  10  12  10  11  12  10  11  9  12

• Put the numbers in numerical order:
• 9  10  10  10  10  11  11  12  12  12  14

• Find the middle number:
• 9  10  10  10  10  11  11  12  12  12  14

The middle number is 11, so the median is 11.

Finding the median with an even number of results

To find the median of the numbers: 5  11  12  4  8  21.

• Put the results in order:
• 4 5  8  11  12  21.

• Find the middle number or numbers:
• 4 5  8  11  12  21.

If there are two central numbers, we need to find their mean.

The median is therefore:

(8 + 11) ÷ 2 = 9.5

Finding the median from a frequency table

• From the table, we can find the total number of CDs:
• 1 + 4 + 2 + 3 + 0 + 1 = 11 CDs

As the results are in a table, they are already ordered for us. In this case, the median is the 6^th result.

Looking at the table we can see that the first 5 CDs have either 9 or 10 tracks. The 6^th CD has 11 tracks. Therefore, this is the median.

If there are n results, the median will be the ^{n + 1}₂^th result.

For example:

For 5 numbers, the median is the (5 + 1) ÷ 2 = 3^rd result

For 6 numbers, the median is the (6 + 1) ÷ 2 = 3.5^th result.

The mode

The mode is the number which occurs most often in a set of data. There can be more than one mode.

Have another look at Kieran's results and find the mode:

10  14  10  12  10  11  12  10  11  9  12

We need to identify the number that appears in this set of numbers the most. Sometimes it helps to list the numbers in numerical order:

9  10  10  10  10  11  11  12  12  12  14

Here we can see that the number 10 occurs most often, so 10 is the mode.

From a frequency table, it is even easier to spot the mode.

There are four CDs with 10 tracks. The other numbers of tracks do not occur four times or more, so 10 is the mode.

Group data and modal classes

When we have a lot of categories, we sometimes need to group the data.

This table shows the shoe sizes of 100 people in a shop:

The groups 10-19, 20-29, 30-39 and 40-49 are called classes.

The class with the most people is 40-49. We therefore call this the modal class.

Measures of average in grouped and continuous data

The mean

We already know how to find the mean from a frequency table. Finding the mean for grouped or continuous data is very similar.

The grouped frequency table shows the number of CDs bought by a class of children in the past year.

• We know that 10 children have bought either 0, 1, 2, 3 or 4 CDs, but we do not know exactly how many each child bought.
• If we assumed that each child bought 4 CDs, it is likely that our estimate of the mean would be too big.
• If we assumed that each child bought 0 CDs, it is likely that our estimate would be too small.
• It therefore seems sensible to use the mid-point of the group and assume that each child bought 2.

Finding the mid-points of the other groups, we get:

The mean is

Remember: This is only an estimate of the mean.

The median

As explained previously, the median is the middle value when the values are arranged in order of size.

As the data has been grouped, we cannot find an exact value for the median, but we can find the class which contains the median.

There are 30 children, so we are looking for the class which contains the (30 + 1) ÷ 2 = 15¹₂th value. The median is therefore within the 5-9 class.

The mode

The mode is the most common value.

We cannot find an exact value for the mode, and therefore give the modal class. The modal class is 5-9.

The range

• Here are Kieran's results again:
• 10  14  10  12  10  11  12  10  11  9  12

Kieran's highest number is 14, and the lowest 9.

Therefore, the range is 14 - 9 = 5.

Remember that to find the range of a set of numbers, you need to:

• Find the lowest number.
• Find the highest number.
• Subtract the lowest number from the highest number.
• The difference between the highest and lowest number is the range.

Comparing distributions

We already know that the mean number of tracks on Kieran's CDs is 11. Deepa also found that the mean number of tracks on her CDs is 11. She therefore came to the conclusion that her results were very similar to Kieran's. Is she correct?

• Here are Kieran's results:
• 10  14  10  12  10  11  12  10  11  9  12

• Here are Deepa's results:
• 2  20  3  17  20  4  5  19  18  2

Although the mean was identical to the mean calculated from Kieran's results, the data is very different. The number of tracks on Deepa's CDs is either very large or very small. Therefore, the range for Deepa's results is also large. From 2 to 20 = 18

Much more information can be gained by giving a measure of central tendency (an average) and a measure of spread (eg, the range).

Now try a Test Bite