Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
KS3 Mathematics D2 Processing data 1 of 42 © Boardworks Ltd 2004 Contents D2 Processing data D2.1 Finding the mode D2.2 Calculating the mean D2.3 Finding the median D2.4 Finding the range D2.5 Calculating statistics 2 of 42 © Boardworks Ltd 2004 Finding the mode The mode or modal value in a set of data is the data value that appears the most often. For example, the number of goals scored by the local football team in the last ten games is: 2, 1, 2, 0, 0, 2, 3, 1, 2, 1. Is it possible to have more than one modal value? Yes The modal score is 2. Is it possible to have no modal value? 3 of 42 Yes © Boardworks Ltd 2004 Finding the mode A dice was thrown ten times. These are the results: What was the modal score? 3 is the modal score because it appears most often. 4 of 42 © Boardworks Ltd 2004 Finding the mode The mode is the only average that can be used for categorical or non-numerical data. For example, 30 pupils are asked how they usually travel to school. The results are shown in a frequency table. Method of travel Frequency Bicycle 6 8 On foot Car 2 Bus 6 Train 3 5 of 42 What is the modal method of travel? Most children travel by foot. Travelling on foot is therefore the modal method of travel. © Boardworks Ltd 2004 Finding the mode from a bar chart This bar chart shows the scores in a science test: Number of pupils 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 Mark out of ten What was the modal score? 6 is the modal score because it has the highest bar. 6 of 42 © Boardworks Ltd 2004 Finding the mode from a pie chart This pie chart shows the favourite food of a sample of people: 23 78 55 18 chocolate fruit vegetables sweets other What was the modal food type? 26 The biggest sector of the pie chart is for chocolate, so this is the modal food type. 7 of 42 © Boardworks Ltd 2004 Finding the mode from a frequency table This frequency table shows the frequency of different length words in a given paragraph of text. Word length 1 Frequency 3 2 3 4 16 12 16 5 7 6 3 7 11 8 6 9 2 10 1 What was the modal word length? We need to look for the word lengths that occur most frequently. For this data there are two modal word lengths: 2 and 4. 8 of 42 © Boardworks Ltd 2004 Finding the modal class for continuous data This grouped frequency table shows the times 50 girls and 50 boys took to complete one lap around a race track. Frequency Time (minutes:seconds) Boys Girls 2:00 ≤ 2:15 3 1 2:15 ≤ 2:30 7 6 2:30 ≤ 2:45 11 10 2:45 ≤ 3:00 13 9 3:15 ≤ 3:30 8 12 3:30 ≤ 3:45 7 10 3:45 ≤ 4:00 1 2 9 of 42 What is the modal class for the girls? What is the modal class for the boys? What is the modal class for the pupils regardless of whether they are a boy or a girl? © Boardworks Ltd 2004 Contents D2 Processing data D2.1 Finding the mode D2.2 Calculating the mean D2.3 Finding the median D2.4 Finding the range D2.5 Calculating statistics 10 of 42 © Boardworks Ltd 2004 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. Sum of values Mean = Number of values For example, the mean of 3, 6, 7, 9 and 9 is 3+6+7+9+9 5 11 of 42 = 34 5 = 6.8 © Boardworks Ltd 2004 The mean 12 of 42 © Boardworks Ltd 2004 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 pupils has so far add up to 78% + 75% + 82% = 235% The mark needed in the fourth test is 320% – 235% = 85% 13 of 42 © Boardworks Ltd 2004 Calculating the mean from a frequency table The following frequency table shows the scores obtained when a dice 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 42 171 171 The mean score = 50 14 of 42 = 3.42 © Boardworks Ltd 2004 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. 15 of 42 © Boardworks Ltd 2004 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. 16 of 42 © Boardworks Ltd 2004 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. 17 of 42 © Boardworks Ltd 2004 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. 18 of 42 © Boardworks Ltd 2004 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 = 19 of 42 1061 = 2.122 500 © Boardworks Ltd 2004 Using an assumed mean We can calculate the mean of a set of data using an assumed mean. For example, 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. For this set of values we can use an assumed mean of 150 cm. 20 of 42 © Boardworks Ltd 2004 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, –8 cm, 5 cm, 18 cm, –5 cm, 2 cm, 7 cm, 0 cm, 6 cm, –12 cm. Next, find the mean of the new set of values. 38 – 27 10 21 of 42 = 11 10 = 1.1 © Boardworks Ltd 2004 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. 22 of 42 © Boardworks Ltd 2004 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. 4) Add this mean to the assumed mean to find the actual mean. Sum of the differences Actual mean = Assumed mean + Total number of values 23 of 42 © Boardworks Ltd 2004 Contents D2 Processing data D2.1 Finding the mode D2.2 Calculating the mean D2.3 Finding the median D2.4 Finding the range D2.5 Calculating statistics 24 of 42 © Boardworks Ltd 2004 Finding the median The median is the middle value of a set of numbers arranged in order. For example, find the median of 10, 7, 9, 12, 7, 8, 6, Write the values in order: 6, 7, 7, 8, 9, 10, 12. The median is the middle value. 25 of 42 © Boardworks Ltd 2004 Finding the median When there is an even number of values, there will be two values in the middle. In this case, we have to find the mean of the two middle values. For example, Find the median of 56, 42, 47, 51, 65 and 43. The values in order are: 42, 43, 47, 51, 56, 65. There are two middle values, 47 and 51. 26 of 42 © Boardworks Ltd 2004 Finding the median To find the number that is half-way between 47 and 51 we can add the two numbers together and divide by 2. 47 + 51 2 = 98 2 = 49 Alternatively, find the difference between 47 and 51 and add half this difference to the lower number. 51 – 47 = 4 ½ of 4 = 2 2 + 47 = 49 The median of 42, 43, 47, 51, 56 and 65 is 49. 27 of 42 © Boardworks Ltd 2004 Find the median 28 of 42 © Boardworks Ltd 2004 Rogue values The median is often used when there is a rogue value – that is, a value that is much smaller or larger than the rest. What is the rogue value in the following data set: 192, 183, 201, 177, 193, 197, 4, 186, 179? The median of this data set is: 4, 177, 179, 183, 186, 192, 193, 197, 201. The median of the data set is not affected by the rogue value, 4. The mean of the data set is 168. This is not representative of the set because it is lower than almost all the data values. 29 of 42 © Boardworks Ltd 2004 Mean or median? Would it be better to use the median or the mean to represent the following data sets? 34.2, 36.8, 29.7, 356, 42.5, 37.1? median 0.4, 0.5, 0.3, 0.8, 0.7, 1.0? mean 892, 954, 1026, 908, 871, 930? mean 3.12, 3.15, 3.23, 9.34, 3.16, 3.20? median 97.85, 95.43, 102.45, 98.02, 97.92, 99.38? mean 87634, 9321, 78265, 83493, 91574, 90046? median 30 of 42 © Boardworks Ltd 2004 Contents D2 Processing data D2.1 Finding the mode D2.2 Calculating the mean D2.3 Finding the median D2.4 Finding the range D2.5 Calculating statistics 31 of 42 © Boardworks Ltd 2004 Finding the range The range of a set of data is a measure of how the data is spread across the distribution. To find the range we subtract the lowest value in the set from the highest value. Range = highest value – lowest value When the range is small it tells us that the values are similar What does it mean if the range is small? in size. When the range is large it tells us that the values vary widely What does it mean if the range is large? in size. 32 of 42 © Boardworks Ltd 2004 Find the range 33 of 42 © Boardworks Ltd 2004 Contents D2 Processing data D2.1 Finding the mode D2.2 Calculating the mean D2.3 Finding the median D2.4 Finding the range D2.5 Calculating statistics 34 of 42 © Boardworks Ltd 2004 Remember the three averages and range S M M I A M E D I A N C L D MODE L L L M RANGE E M R S MEAN O G T D N E DIVIDE S T 35 of 42 © Boardworks Ltd 2004 The three averages and range There are three different types of average: MEAN MEDIAN MODE sum of values number of values middle value most common The range is not an average, but tells you how the data is spread out: RANGE largest value – smallest value 36 of 42 © Boardworks Ltd 2004 The three averages Each type of average has its purpose and sometimes one is preferable to an other. 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. The mean is the most difficult to calculate but takes into account all the values in the data set. 37 of 42 © Boardworks Ltd 2004 Find the mean, median and range 38 of 42 © Boardworks Ltd 2004 Find the missing value 39 of 42 © Boardworks Ltd 2004 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. The median is bigger than the mean. The mean and the median are the same. 40 of 42 © Boardworks Ltd 2004 Stem-and-leaf diagrams Sometimes data is arranged in a stem-and-and leaf diagram. For example, this stem-and-leaf diagram shows the marks scored by 21 pupils in a maths test. stem = tens leaves = units Find the median, mode and range for the data. 0 6 7 9 1 4 5 5 8 2 0 1 3 5 6 6 There are 21 data values so the median will be the 25 . 11th value, that is ___ 3 0 2 2 2 5 8 The mode is ___ 32 . 4 0 0 The range is 40 – 6, 34 . which is ___ 41 of 42 © Boardworks Ltd 2004 Stem-and-leaf diagrams 42 of 42 © Boardworks Ltd 2004