Year 8 Mathematics Averages http://www.mathsisfun.com/fractions.html Averages

Learning Intentions You should be able to: Calculate Mean, Median, Mode and Range Interpret results Draw and read from a Bar chart,

The Average An Average is a single value that best represents a data set. When you receive your examinations results you want to compare your marks with everyone else’s mark. To do this you use the Year or Class average. This is a single value that best represents how everyone else performed.

My Average Sometime you may wish to find how well you have done overall. You need to find the average of all your marks. For example, you may have received the following scores: Mathematics 70% French 80% Geography 55% History 75% So how did you do?

Calculating the Average We calculate the average by adding all the scores and dividing by the number of subjects. Average = total score/number of scores = 70 + 80 + 55 + 75 /4 = 280/4 = 70%

What does it Mean? The average you have just calculated is called the Mean. It’s the average that we use most often. You need to remember the formula: Mean = total score/number of scores

Is that all? Imagine that you have had the same £4 per week pocket money for a long time. You are now in Year 8 and think that a Secondary school pupil should have more. You ask some friends and they get £1, £3, £3, £3 and £15 per week. You calculate the mean and get £_______. What do you do?

The Mode The mean is £5- so you feel you should get £1 more! However, when you speak to you parents, they say that the average is £3! They have used the Mode. The mode is an average that looks at the most common value. Which average do you think represents the data set best?

I’m tall! Sometimes when playing sport, the organiser divides people into groups depending on their height. Everyone is put in order and the tallest people form one group and the smallest people form another group. The person in the middle between the two groups has the average height. Half the group are smaller and half are taller.

The Median The Median is the number in the middle – but you need to put the numbers in order! Who is in the middle? Put these heights in order and find the middle (median) height. 134cm, 174cm, 152cm, 165cm, 157cm. The median = _______ .

What if there is no Middle? Sometimes when you put the numbers in order there is no number in the middle. For example find the median of these numbers: 1, 2, 3, 4, 5, 6 In these cases, you add the two middle numbers and divide by 2. Median = 3 + 4 /2 = 3.5

Spread Out! Sometimes we need to consider how the data is spread. Consider these two sets of score: 80%, 85%, 80%, 81%, 87%, 85% 70%, 97%, 78%, 85%, 92%, 88% The scores for pupils in the first group are much closer together than those in the second group.

The Range Pupils who are placed in groups depending on their ability might give the first set of results. Pupils with a much wider range of abilities would give the second set of results. The Range is calculated by subtracting the smallest score from the largest score. For 1st set, Range = 87% - 80% = 7% For 2nd set, Range = 97% - 70% = 27%

So who is better? In the previous example, the means and ranges for the data sets are: Set 1: Mean = 83%, Range = 7% Set 2: Mean = 85%, Range = 27% The first group are all of the same ability (range), but tend to be less able than the second group (mean). The second group tend to be more able (mean), but there is wider ability levels (range).

Summarising Data Sometimes when we have a large number of data values, we record the data in a table. This table is called a Frequency Table. For example, These are the shoe sizes for pupils in 9A. 3 7 4 5 8 9 6

Frequency Table We create the frequency table as shown. Don’t forget a total row! x Tally Frequency (f) 3 IIII III 8 4 IIII II 7 5 III 6 II 2 9 Total 28

Calculating the Mean We can calculate the mean from the frequency table. Remember we need to add all the values! There are seven 3’s, seven 4’s, three 5’s, … It’s easier to add an extra column. Mean = _____ x Frequency (f) Total (fx) 3 8 24 4 7 28 5 15 6 18 21 2 16 9 Total 140

Finding the Mode We can also find the mode from a frequency table. Look again at the previous table. The mode is the most common shoe size. The mode is size 3. x Frequency (f) 3 8 4 7 5 6 2 9 Total 28

Finding the Median The frequency table naturally puts the values in order. Look again at the frequency table. There were 28 pupils, so the median is the mean of the 14th and 15th value. The 14th value is 4 and the 15th value is 4, so the median is 4. x Frequency (f) 3 8 4 7 5 6 2 9 Total 28

Illustrating the Results Sometimes we want to use a diagram to show the results. The height of the bar gives the frequency. For example, these are the colours of cars in the car park. Draw a bar graph. Notice: Chart Title Axis labels Gaps between bars Sometimes a legend is included Colour Frequency (f) Blue 2 Red 3 White 4 Black Total 12

Piece of the Pie Pie charts are also used to display the data. This is a circular diagram. Each data item is given a sector of the circle. The angle spanned by the sector gives the frequency.

Illustrating the Results For example, these are the colours of cars in the car park. Draw a pie chart. We need to calculate the angle to use. Notice: Chart Title Axis labels Gaps between bars Sometimes a legend is included Colour Frequency (f) Angle Blue 2 60 Red 3 90 White 4 120 Black Total 12 360