Maths- Averages, Charts, Cumulative Frequency, Box and Histograms

Question 1

Mean, mode and median

Answer 1

o draw conclusions from data you compare summary values, a measure of average and a measure of spread. An average indicates the typical value of a set of data. Mean, median and mode are all types of average. Range is a measure of spread.

Question 2

The mean

Answer 2

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.

Question 3

The median

Answer 3

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.

Question 4

Finding the median with an odd number of results

Answer 4

Using the same example, find the median number of tracks on Kieran's CDs.
Kieran's results were:
9 13 9 11 9 13 11 9 10 8 11
Put the numbers in numerical order:
8 9 9 9 9 10 11 11 11 13 13
Find the middle number:
8 9 9 9 9 10 11 11 11 13 13
The middle number is 10, so the median is 10.

Question 5

Finding the median with an even number of results

Answer 5

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

Question 6

The mode

Answer 6

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

Question 7

The Range

Answer 7

Highest number, minus the lowest number gives you the range of results.

Question 8

bar charts

Answer 8

Bar charts or bar graphs represent data as vertical blocks or columns.
The X axis shows what type of data each column represents, and the Y axis shows a value for that type of data. For example, in a rainfall graph, each column on the X axis represents a month of the year, with the height of each column on the Y axis showing the amount of rainfall in that month.

Question 9

pie charts

Answer 9

Pie charts are used for showing how something breaks down into its constituent parts.
Pie charts are usually drawn so the biggest piece of pie comes first (starting at 12 o’clock).
The ‘others’ section usually goes last (coming up to 12 o’clock).
If each piece of pie has its percentage written in, it’s easier for the reader to work out the exact proportions.

Question 10

line graphs

Answer 10

Different types of graph are used to represent statistical data. The most common types are line graphs, bar graphs and pie charts.
Most graphs have two axes: the X axis is horizontal (across the bottom) while the Y axis is vertical (up the left side). The two axes each represent a different set of data. Line graphs are drawn by plotting points by their X and Y coordinates, then joining them together or drawing a line through the middle.

Question 11

Cumulative frequency

Answer 11

The cumulative frequency is obtained by adding up the frequencies as you go along, to give a ‘running total’.

Question 12

Drawing a cumulative frequency diagram

Answer 12

Before drawing the cumulative frequency diagram, we need to work out the cumulative frequencies. This is done by adding the frequencies in turn.

Question 13

Example:

Answer 13

Length  Frequency	Cumulative Frequency
21-24	3	        3
25-28	7	        10 (= 3 + 7)
29-32	12	        22 (= 3 + 7 + 12)
33-36	6	        28 (= 3 + 7 + 12 + 6)
37-40	4	        32 (= 3 + 7 + 12 + 6 + 4)

Question 14

Box and whisker plots

Answer 14

A box and whisker plot is used to display information about the range, the median and the quartiles. It is usually drawn alongside a number line

Question 15

Example

Answer 15

The oldest person in Mathsminster is 90. The youngest person is 15.
The median age of the residents is 44, the lower quartile is 25, and the upper quartile is 67.
Represent this information with a box-and-whisker plot.

Question 16

Histograms

Answer 16

The following table shows the ages of 25 children on a school bus:

Age	Frequency
5-10	        6
11-15       15
16-17	 4
> 17	         0
If we are going to draw a histogram to represent the data, we first need to find the class boundaries. In this case they are 5, 11, 16 and 18. The class widths are therefore 6, 5 and 2.

Question 17

Remember that in a bar chart the height of the bar represents the frequency. It is therefore correct to label the vertical axis ‘frequency’.
However, as in a histogram, it is the area which represents the frequency.
It would therefore be incorrect to label the vertical axis ‘frequency’ and the label should be ‘frequency density’.

Answer 17

So we know that Area = frequency = Frequency density x class width hence:
Frequency density = frequency ÷ class width