Averages
To draw conclusions from data, it is useful to calculate averageA value to best represent a set of data. There are three type of average - the mean, the median and the mode..
An average indicates the typical value of a set of data and the main types are meanThe average., medianThe median is the value of the middle item of data when all the data is arranged in order. and modeThe mode is the item which occurs most often..
You can also get more information from your data by giving a measure of spread.
Determining averages (mean, median and mode)
As part of a school project, Kieran is asked to write down the number of tracks on each of his albums. His results are as follows:
10, 14, 10, 12, 11, 10, 11, 12, 10, 11, 9 and 12.
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, 11, 10, 11, 12, 10, 11, 9 and 12.
To calculate the mean, add the numbers together and divide the total by the amount of numbers.
The mean for this example is:
\(\frac{{10 + 14 + 10 + 12 + 11 + 10 + 11 + 12 + 10 + 11 + 9 + 2}}{{11}} = \frac{{132}}{{11}} = 11\)
Question
A joiner bought 7 packets of nails. The number of nails in each packet was as follows:
8, 7, 6, 7, 9, 8, 7
Calculate the mean number of nails per packet. (Give your answer to 2 decimal places)
Total nails = 8 + 7 + 6 + 7 + 9 + 8 + 7 = 52
Number of packets = 7
Mean = \(\frac{52}{7}\)= 7.43
Finding the mean from a frequency table
Putting Kieran's results (data) into a frequency table looks like this.
| Number of tracks on album | Number of albums (Frequency) |
| 9 | 1 |
| 10 | 4 |
| 11 | 3 |
| 12 | 3 |
| 13 | 0 |
| 14 | 1 |
| Number of tracks on album | 9 |
|---|---|
| Number of albums (Frequency) | 1 |
| Number of tracks on album | 10 |
|---|---|
| Number of albums (Frequency) | 4 |
| Number of tracks on album | 11 |
|---|---|
| Number of albums (Frequency) | 3 |
| Number of tracks on album | 12 |
|---|---|
| Number of albums (Frequency) | 3 |
| Number of tracks on album | 13 |
|---|---|
| Number of albums (Frequency) | 0 |
| Number of tracks on album | 14 |
|---|---|
| Number of albums (Frequency) | 1 |
We could have found the mean of his results by using this method
| Number of tracks on album | Frequency | Tracks x Frequency |
| 9 | 1 | 9 |
| 10 | 4 | 40 |
| 11 | 3 | 33 |
| 12 | 3 | 36 |
| 13 | 0 | 0 |
| 14 | 1 | 14 |
| Total = 12 | Total = 132 |
| Number of tracks on album | 9 |
|---|---|
| Frequency | 1 |
| Tracks x Frequency | 9 |
| Number of tracks on album | 10 |
|---|---|
| Frequency | 4 |
| Tracks x Frequency | 40 |
| Number of tracks on album | 11 |
|---|---|
| Frequency | 3 |
| Tracks x Frequency | 33 |
| Number of tracks on album | 12 |
|---|---|
| Frequency | 3 |
| Tracks x Frequency | 36 |
| Number of tracks on album | 13 |
|---|---|
| Frequency | 0 |
| Tracks x Frequency | 0 |
| Number of tracks on album | 14 |
|---|---|
| Frequency | 1 |
| Tracks x Frequency | 14 |
| Number of tracks on album | |
|---|---|
| Frequency | Total = 12 |
| Tracks x Frequency | Total = 132 |
When we divide these two totals as follows:
Mean = \(\frac{132}{12}\) = 11 tracks.
