S4 Mean and calculating statistics Boardworks KS3 Maths 2009 S4 Mean and calculating statistics S4 Mean and calculating statistics This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics S4.1 Calculating the mean

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics The mean The mean is the most commonly used average. To calculate the mean of a set of values we add together the values and divide by the total number of values. Number of values Sum of values Mean = For example, the mean of 3, 6, 7, 9 and 9 is 3 + 6 + 7 + 9 + 9 5 5 34 = = 6.8

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics The mean Encourage pupils to calculate the mean themselves. After that let them check their results with the correct value calculated by the activity.

Problems involving the mean Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Problems involving the mean A pupil scores 78%, 75% and 82% in three tests. What must she score in the fourth test to get an overall mean of 80%? To get a mean of 80% the four marks must add up to: 4 × 80% = 320%. The three marks that the pupil has so far add up to: 78% + 75% + 82% = 235%. Photo credit: © Shutterstock 2009, Monkey Business Images Start by asking students what other four scores would give overall mean of 80%. Explain why any four scores must add up to 320% to give a mean of 80%. Encourage pupils to check the answer by adding 78, 75, 82 and 85 and dividing by four. The mark needed in the fourth test is: 320% – 235% = 85%.

Calculating the mean from a frequency table Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating the mean from a frequency table The following frequency table shows the scores obtained when a die is thrown 50 times. What is the mean score? Score 1 2 3 4 5 6 Total Frequency 8 11 6 9 9 7 50 Score × Frequency 8 22 18 36 45 42 171 Explain that to find the mean score we need to find the total score altogether. A 1 was scored 8 times and so we can find the score obtained by throwing 1s by multiplying 8 × 1. A 2 was scored 11 times and so we can find the score obtained by throwing 2s by finding 11 × 2. Conclude that we need to find the score × the frequency for each score. Show how this can be done by revealing the yellow row in the table. We can then record the total number of throws and the total score in the blue column. The mean is found by dividing these totals. 171 50 The mean score = = 3.42

Calculating the mean using a spreadsheet Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating the mean using a spreadsheet When processing large amounts of data it is often helpful to use a spreadsheet to help us calculate the mean. For example, 500 households were asked how many children under the age of 16 lived in the home. The results were collected in a spreadsheet. Discuss how a spreadsheet can be used to find the mean for larger amounts of data.

Calculating the mean using a spreadsheet Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating the mean using a spreadsheet The total number of households is found by entering =SUM(B2:J2) in cell K2 as follows: Pressing enter shows the number of households.

Calculating the mean using a spreadsheet Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating the mean using a spreadsheet The total number of children in households with no children is found by entering =B1*B2 into cell B3. The total number of children in households with one child is found by entering =C1*C2 into cell C3. Formulas can also be created by dragging the formula from B3 across the desired area using the fill handle.

Calculating the mean using a spreadsheet Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating the mean using a spreadsheet This can be repeated along the row to find the total number of children in each type of household. To find the total number of children altogether enter =SUM(B3:J3) in cell K3.

Calculating the mean using a spreadsheet Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating the mean using a spreadsheet The mean number of children in each household is now found by dividing the number in cell K3 by the number in cell K2. Mean number of children = 1061 500 = 2.122 Explain that it is a good idea to create a cell containing the formula =K3/K2. This cell will calculate the mean number of children. The advantage of entering this as a formula is that if any of the values in the table are changed, the mean will be recalculated automatically. Point out that the mean provides an answer including decimals and discuss what this means in context.

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Using an assumed mean We can calculate the mean of a set of data using an assumed mean. Suppose the heights of ten year 8 pupils are as follows: 148 cm, 155 cm, 145 cm, 157 cm, 156 cm, 142 cm, 168 cm, 152 cm, 150 cm, 138 cm. To find the mean of these values using an assumed mean we start by making a guess at what the mean might be. This is the assumed mean. Emphasize that the assumed mean is going to be subtracted from each of the values in the set of data. It makes sense, therefore, to choose a value that is easy to subtract, such as a value that has been rounded to the nearest 10. Point out, that it doesn’t really matter what this value is as long as it is easy to subtract and the resulting differences are easy to add. The aim is to make the values smaller so that we can calculate the mean mentally. For this set of values we can use an assumed mean of 150 cm.

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Using an assumed mean Subtract the assumed mean from each of the data values. 148 cm, 155 cm, 145 cm, 157 cm, 156 cm, 142 cm, 168 cm, 152 cm, 150 cm, 138 cm. – 150 cm –2 cm, 5 cm, –5 cm, 7 cm, 6 cm, –8 cm, 18 cm, 2 cm, 0 cm, –12 cm. Next, find the mean of the new set of values. Point out that when we add together lots of numbers where some are positive and some are negative, it is often easier to add the positive numbers together and then add the negative numbers together. That way, we only have a single subtraction to do at the end of the calculation. If we add together the positive values we get 38 and if we add the negative values together we get –27. The new set of values therefore have a sum of 11. 38 – 27 10 10 11 = = 1.1

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Using an assumed mean To find the actual mean of the heights, add this value to the assumed mean. 150 + 1.1 = 151.1 assumed mean mean of the differences actual mean So, the actual mean of the heights is 151.1 cm. This method is often used to find the mean of numbers that are large or written to a large number of decimal places. We can use this method to find the mean mentally, using jottings.

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Using an assumed mean In summary, to find the mean of a set of values using an assumed mean follow these steps: 1) Assume the mean of the values. 2) Subtract this assumed mean from each of the values. 3) Find the mean of the new values. Add this mean to the assumed mean to find the actual mean. Sum of the differences Total number of values Actual mean = Assumed mean +

S4.2 Calculating statistics Boardworks KS3 Maths 2009 S4 Mean and calculating statistics S4.2 Calculating statistics

Remember the three averages and range Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Remember the three averages and range R A N G E L A R G E S T M M E D I A N M I D D L E M O D E C O M N M E A N A D D I V I D E

The three averages and range Boardworks KS3 Maths 2009 S4 Mean and calculating statistics The three averages and range There are three different types of average: MEAN MEDIAN middle value MODE most common sum of values number of values The range is not an average, but tells you how the data is spread out: RANGE largest value – smallest value

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics The three averages Each type of average has its purpose and sometimes one is preferable to another. The mode is easy to find and it eliminates some of the effects of extreme values. It is the only type of average that can be used for categorical (non-numerical) data. The median is also fairly easy to find and has the advantage of being hardly affected by rogue values or skewed data. Discuss that advantages and disadvantages of each type of average. The mean is the most difficult to calculate but takes into account all the values in the data set.

Find the mean, median and range Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Find the mean, median and range Use the activity to practise finding the mean, median and range. If there is a modal value for a particular data set, ask pupils to tell you what this is.

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Find the missing value Pupils must use the information given to find the missing value.

Calculating statistics Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Calculating statistics Look at the values on these five cards: 2 4 5 8 11 Choose three cards so that: The mean is bigger than the median. Establish that for the mean to be bigger than the median, the first two values must be much lower than the third value. For example, the mean of 2, 5 and 11 is 6 and the median is 5. For the median to be bigger than the mean the second two values must be much greater than the first. For example, the mean of 2, 8 and 11 is 7 and the median is 8. For the median and the mean to be the same, the difference between the first value and the second value must be the same as the difference between second value and the third value. For example, the mean and the median of 2, 5 and 8 is 5. Make the activity more challenging by choosing decimal values for the cards, changing the number of cards or asking pupils to choose four cards that satisfy the required conditions. The median is bigger than the mean. The mean and the median are the same.

Stem-and-leaf diagrams Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Stem-and-leaf diagrams Sometimes data is arranged in a stem-and-leaf diagram. The below stem-and-leaf diagram shows the marks scored by 21 pupils in a maths test. Find the median, mode and range for the data. stem = tens leaves = units 1 2 3 4 6 7 9 There are 21 data values so the median will be the 11th value, that is ___ . 4 5 5 8 1 3 5 5 6 6 25 2 2 2 2 5 8 Start by explaining how to read the stem-and-leaf diagram. The scores shown in the diagram are 6, 7, 9, 14, 15, 15, 18, 20, 21, 23, 25, 26, 26, 30, 32, 32, 32, 35, 38, 40 and 40. Ask pupils what they think the test might have been out of. The values used for the stem and leaves depend on the data. For example, if we were showing times in minutes and seconds, we could use the stem to show the minutes and the leaves to show the seconds. If we were showing lengths between 3.0 and 8.0 written to one decimal place, we could use the stem to show the whole number of centimetres and the leaves to show the tenths (or millimetres). Discuss how to find the median, mode and range. The mode is ___ . 32 The range is 40 – 6, which is ___ . 34

Stem-and-leaf diagrams Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Stem-and-leaf diagrams Calculate the median, mode and range for each stem-and-leaf diagram.

Boardworks KS3 Maths 2009 S4 Mean and calculating statistics S4.3 Comparing data

Comparing distributions Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Comparing distributions The distribution of a set of data describes how the data is spread out. Two distributions can be compared using one of the three averages and the range. For example, the number of cars sold by two salesmen each day for a week is shown below. Matt 5 7 6 5 7 8 6 Photo credit: © Shutterstock 2009, Szocs Jozsef Ask pupils who they think is the best and why? Jamie 3 6 4 8 12 9 8 Who is the better salesman?

Comparing distributions Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Comparing distributions Matt 5 7 6 5 7 8 6 Jamie 3 6 4 8 12 9 8 To decide which salesman is best, let’s compare the mean number cars sold by each one. Matt: 5 + 7 + 6 + 5 + 7 + 8 + 6 7 = 44 7 Mean = = 6.3 (to 1 d.p.) Jamie: 3 + 6 + 4 + 8 + 12 + 9 + 8 7 = 50 7 Mean = = 7.1 (to 1 d.p.) This tells us that, on average, Jamie sold more cars each day.

Comparing distributions Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Comparing distributions Matt 5 7 6 5 7 8 6 Jamie 3 6 4 8 12 9 8 Now let’s compare the range for each salesman. Matt: Range = 8 – 5 = 3 Jamie: Range = 12 – 3 = 9 The range for the number of cars sold each day is smaller for Matt. This means that he is a more consistent or reliable salesman. A small range means that the values are less likely to fluctuate. An employer may prefer a salesman who can be relied upon to sell a minimum number of cars each day. Although, Matt sold 12 cars on one day, he also had two ‘bad days’ where he only sold 3 and 4 cars respectively. Summarize to pupils that we could argue that either one of them is better. The important point is to provide a reason for the choice using the data. We could argue that Jamie is better because he sells more on average, or that Matt is better because he is more consistent.

Comparing the shape of distributions Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Comparing the shape of distributions We can compare distributions by looking at the shape of their graphs. This distribution is symmetrical (or normal). This distribution is skewed to the left. This distribution is skewed to the right. This distribution is random. Ask pupils how they think the shapes of these distributions would affect the mean for each one. Ask them if they think one of the other averages would be more appropriate for skewed data.

Comparing the shape of distributions Boardworks KS3 Maths 2009 S4 Mean and calculating statistics Comparing the shape of distributions Four groups of pupils sat the same maths test. These graphs show the results. Group B 1-10 11-20 21-30 31-40 41-50 Frequency Group C 1-10 11-20 21-30 31-40 41-50 Frequency Group D 1-10 11-20 21-30 31-40 41-50 Frequency Group A Frequency 1-10 11-20 21-30 31-40 41-50 One of the groups is a top set, one is a middle set, one is a bottom set and one is a mixed ability group. Ask pupils what sort of data is shown in the graphs (grouped discrete data). Discuss how we can use the shape of each graph to establish the ability of each group. Ask pupils what they would expect the mean percentage for each group to be. Group A has a similar number of pupils for each interval. This is most likely to be the mixed ability group. The mean score for this group will be around 50%. Group B has more pupils gaining lower scores. The data is skewed to the left. This is most likely to be the bottom set. The mean score for this group will be around 25%. Group C has more pupils gaining higher scores. The data is skewed to the right. This is most likely to be the top set. The mean score for this group will be around 75%. Group D has more pupils gaining average scores. This is most likely to be the middle set. The mean score for this group will be around 50%. Notice that the mean scores for group A and group D are likely to be very similar although their distributions are very different. They may also have a similar range. This demonstrates that finding the average and the range is sometimes not sufficient to get a clear idea of how the data is distributed. This is why looking at the shape of a distribution is important. Use the shapes of the distribution to decide which group is which giving reasons for your choice.