# 15-Apr-15Created by Mr. Lafferty1 Statistics www.mathsrevision.com Mode, Mean, Median and Range Semi-Interquartile Range ( SIQR ) Nat 5 Quartiles Boxplots.

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15-Apr-15Created by Mr. Lafferty1 Statistics www.mathsrevision.com Mode, Mean, Median and Range Semi-Interquartile Range ( SIQR ) Nat 5 Quartiles Boxplots – Five Figure Summary Full Standard Deviation Sample Standard Deviation Exam questions

Nat 5 15-Apr-15Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com xoxo 42 o

Nat 5 15-Apr-15Created by Mr Lafferty Maths Dept Statistics Learning Intention Success Criteria 1.Understand the terms 1.Understand the terms mean, range, median and mode. 1.We are revising the terms mean, median, mode and range. 2.To be able to calculate mean, range, mode and median. www.mathsrevision.com Averages

15-Apr-15Created by Mr Lafferty Maths Dept 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: What is the modal score? Is it possible to have more than one modal value? Is it possible to have no modal value? Yes 2,1,2,0, 2,3,1,2,1.2,1,2,0, 2,3,1,2,1. 2. Statistics www.mathsrevision.com Nat 5

15-Apr-15Created by Mr Lafferty Maths Dept 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. For example, the mean of 3, 6, 7, 9 and 9 is Mean = Sum of values Number of values Statistics www.mathsrevision.com Nat 5

15-Apr-15Created by Mr Lafferty Maths Dept 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, 8,9,10,12. The median is the middle value. Statistics www.mathsrevision.com Nat 5

15-Apr-15Created by Mr Lafferty Maths Dept Finding the median When there is an even number of values, there will be two values in the middle. For example, Find the median of 56, 42, 47, 51, 65 and 43. The values in order are: There are two middle values, 47 and 51. 42,43,47,51,56,65. 47 + 51 2 = 98 2 = 49 Statistics www.mathsrevision.com Nat 5

15-Apr-15Created by Mr Lafferty Maths Dept 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 large; the values vary widely in size. When the range is small; the values are similar in size. Statistics www.mathsrevision.com Nat 5

15-Apr-15Created by Mr Lafferty Maths Dept Here are the high jump scores for two girls in metres. Joanna1.621.411.351.201.15 Kirsty1.591.451.411.301.30 Find the range for each girl’s results and use this to find out who is consistently better. Joanna’s range = 1.62 – 1.15 = 0.47 Kirsty’s range = 1.59 – 1.30 = 0.29 The range Statistics www.mathsrevision.com Nat 5 Kirsty is consistently better !

15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com No of Bulbs (c) Freq.(f) Example : This table shows the number of light bulbs used in people’s living rooms 7 5 1 20 2 5 1 2 3 4 5 Totals Frequency Tables Working Out the Mean Adding a third column to this table will help us find the total number of bulbs and the ‘Mean’. 7 x 1 = 7 5 x 3 = 15 1 x 5 = 5 2 x 4 = 8 5 x 2 = 10 45 (f) x (B) Nat 5

15-Apr-15Created by Mr Lafferty Maths Dept Now try N5 TJ Ex 11.1 Ch11 (page 104) www.mathsrevision.com Statistics Nat 5 Averages

15-Apr-15Created by Mr. Lafferty12 www.mathsrevision.com Lesson Starter Q1. Q2.Calculate sin 90 o Q3.Factorise 5y 2 – 10y Q4. A circle is divided into 10 equal pieces. Find the arc length of one piece of the circle if the radius is 5cm. Nat 5

15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.We are learning about Quartiles. 1.Understand the term Quartile. Quartiles Nat 5 2.Be able to calculate the Quartiles for a set of data.

15-Apr-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Quartiles :Splits a dataset into 4 equal lengths. Nat 5 Quartiles 25% Q 1 Q 2 Q 3 25%50%75% Median

15-Apr-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Nat 5 Quartiles Note : Dividing the number of values in the dataset by 4 and looking at the remainder helps to identify quartiles. R1 means to can simply pick out Q 2 (Median) R2 means to can simply pick out Q 1 and Q 3 R3 means to can simply pick out Q 1, Q 2 and Q 3 R0 means you need calculate them all

Quartiles 15-Apr-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Example 1 :For a list of 9 numbers find the SIQR 3, 3, 7, 8, 10, 9, 1, 5, 9 2 numbers2 numbers2 numbers 2 numbers Q1Q2Q3 The quartiles are Q 1 :the 2 nd and 3 rd numbers Q 2 :the 5 th number Q 3 :the 7 th and 8 th number. Nat 5 1 No. 3 7 9 9 ÷ 4 = 2R1 1 3 3 5 7 8 9 9 10 Semi-interquartile Range (SIQR) = ( Q 3 – Q 1 ) ÷ 2 = ( 9 – 3 ) ÷ 2 = 3

Quartiles 15-Apr-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Example 3 :For the ordered list find the SIQR. 3, 6, 2, 10, 12, 3, 4 1 number1 number1 number1 number Q1Q2Q3 The quartiles are Q 1 :the 2 nd number Q 2 :the 4 th number Q 3 :the 6 th number. 7 ÷ 4 = 1R3 Nat 5 3 4 10 Semi-interquartile Range (SIQR) = ( Q 3 – Q 1 ) ÷ 2 = ( 10 – 3 ) ÷ 2 = 3.5 2 3 3 4 6 10 12

Nat 5 15-Apr-15Created by Mr Lafferty Maths Dept Now try N5 TJ Ex 11.2 Ch11 (page 106) www.mathsrevision.com Statistics Averages

15-Apr-15Created by Mr. Lafferty19 www.mathsrevision.com Lesson Starter In pairs you have 3 minutes to explain the various steps of factorising. Nat 5

15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.We are learning about Semi-Interquartile Range. 1.Understand the term Semi- Interquartile Range. Semi-Interquartile Range Nat 5 2.Be able to calculate the SIQR.

Nat 5 The range is not a good measure of spread because one extreme, (very high or very low value can have a big effect). Another measure of spread is called the Semi - Interquartile Range and is generally a better measure of spread because it is not affected by extreme values. Inter-Quartile Range www.mathsrevision.com

Upper Quartile = 10 Q3Q3 Lower Quartile = 4 Q1Q1 Median = 8 Q2Q2 3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15, Finding the Semi-Interquartile range. 6, 3, 9, 8, 4, 10, 8, 4, 15, 8, 10 Order the data Inter- Quartile Range = (10 - 4)/2 = 3 Example 1: Find the median and quartiles for the data below.

12, 6, 4, 9, 8, 4, 9, 8, 5, 9, 8, 10 4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12 Order the data Inter- Quartile Range = (9 - 5½) = 1¾ Example 2: Find the median and quartiles for the data below. Lower Quartile = 5½ Q1Q1 Upper Quartile = 9 Q3Q3 Median = 8 Q2Q2 Finding the Semi-Interquartile range.

Nat 5 15-Apr-15Created by Mr Lafferty Maths Dept Now try N5 TJ Ex 11.3 Ch11 (page 108) www.mathsrevision.com Statistics

15-Apr-15Created by Mr. Lafferty25 www.mathsrevision.com Lesson Starter In pairs you have 3 minutes to come up with questions on Straight Line Theory ( Remember you needed to know the answers to the questions ) Nat 5

15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.We are learning about Boxplots and five figure summary. 1.Calculate five figure summary. Boxplots ( 5 figure Summary) Nat 5 2.Be able to construct a boxplot.

456789101112 Median Lower Quartile Upper Quartile Lowest Value Highest Value Box Whisker 130140150160170180190 Boys Girls cm Box and Whisker Diagrams. Box plots are useful for comparing two or more sets of data like that shown below for heights of boys and girls in a class. Anatomy of a Box and Whisker Diagram.

Lower Quartile = 5½ Q1Q1 Upper Quartile = 9 Q3Q3 Median = 8 Q2Q2 456789101112 4, 4, 5, 6, 8, 8, 8, 9, 9, 9, 10, 12 Example 1: Draw a Box plot for the data below Drawing a Box Plot.

Upper Quartile = 10 Q3Q3 Lower Quartile = 4 Q1Q1 Median = 8 Q2Q2 3, 4, 4, 6, 8, 8, 8, 9, 10, 10, 15, Example 2: Draw a Box plot for the data below Drawing a Box Plot. 34567891011 1213 1415

Upper Quartile = 180 QuQu Lower Quartile = 158 QLQL Median = 171 Q2Q2 Question: Stuart recorded the heights in cm of boys in his class as shown below. Draw a box plot for this data. Drawing a Box Plot. 137, 148, 155, 158, 165, 166, 166, 171, 171, 173, 175, 180, 184, 186, 186 130140150160170180190cm

2. The boys are taller on average. Question: Gemma recorded the heights in cm of girls in the same class and constructed a box plot from the data. The box plots for both boys and girls are shown below. Use the box plots to choose some correct statements comparing heights of boys and girls in the class. Justify your answers. Drawing a Box Plot. 130140150160170180190 Boys Girls cm 1. The girls are taller on average. 3. The girls show less variability in height. 4. The boys show less variability in height. 5. The smallest person is a girl 6. The tallest person is a boy

Nat 5 15-Apr-15Created by Mr Lafferty Maths Dept Now try N5 TJ Ex 11.4 Ch11 (page 109) www.mathsrevision.com Statistics

15-Apr-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5

15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know the term Standard Deviation. 1. We are learning the term Standard Deviation for a collection of data. 2.Calculate the Standard Deviation for a collection of data. Nat 5 Standard Deviation For a FULL set of Data

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Standard Deviation For a FULL set of Data The range measures spread. Unfortunately any big change in either the largest value or smallest score will mean a big change in the range, even though only one number may have changed. The semi-interquartile range is less sensitive to a single number changing but again it is only really based on two of the score.

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Standard Deviation For a FULL set of Data A measure of spread which uses all the data is the Standard Deviation The deviation of a score is how much the score differs from the mean.

Nat 5 ScoreDeviation(Deviation) 2 70 72 75 78 80 Totals375 Example 1 :Find the standard deviation of these five scores 70, 72, 75, 78, 80. Standard Deviation For a FULL set of Data Step 1 : Find the mean 375 ÷ 5 = 75 Step 3 : (Deviation) 2 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com -5 -3 0 3 5 0 25 9 0 9 25 68 Step 2 : Score - Mean Step 4 : Mean square deviation 68 ÷ 5 = 13.6 Step 5 : Take the square root of step 4 √13.6 = 3.7 Standard Deviation is 3.7 (to 1d.p.)

Nat 5 Example 2 :Find the standard deviation of these six amounts of money £12, £18, £27, £36, £37, £50. Standard Deviation For a FULL set of Data Step 1 : Find the mean 180 ÷ 6 = 30 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Step 2 : Score - Mean Step 3 : (Deviation) 2 Step 4 : Mean square deviation 962 ÷ 6 = 160.33 ScoreDeviation(Deviation) 2 12 18 27 36 37 50 Totals180 -18 -12 -3 6 7 20 324 144 9 36 49 400 0 962 Step 5 : Take the square root of step 4 √160.33 = 12.7 (to 1d.p.) Standard Deviation is £12.70

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Standard Deviation For a FULL set of Data When Standard Deviation is LOW it means the data values are close to the MEAN. When Standard Deviation is HIGH it means the data values are spread out from the MEAN. MeanMean

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. Now try N5 TJ Ex 11.5 Q1 & Q2 Ch11 (page 111) www.mathsrevision.com Standard Deviation For a FULL set of Data

15-Apr-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Nat 5 In pairs you have 6 mins to write down everything you know about the circle theory. Come up with a circle type of question you could be asked at National 5 Level.

15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1. We are learning how to calculate the Sample Standard deviation for a sample of data. Standard Deviation For a Sample of Data Standard deviation Nat 5 1.Know the term Sample Standard Deviation. 2.Calculate the Sample Standard Deviation for a collection of data.

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Standard Deviation For a Sample of Data In real life situations it is normal to work with a sample of data ( survey / questionnaire ). We can use two formulae to calculate the sample deviation. s = standard deviation n = number in sample ∑ = The sum of x = sample mean We will use this version because it is easier to use in practice !

Nat 5 Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76. 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Standard Deviation For a Sample of Data Heart rate (x)x2x2 70 72 73 74 75 76 Totals 4900 5184 5329 5476 5625 5776 5776 5776 ∑x 2 = 43842 ∑x = 592 Step 2 : Square all the values and find the total Step 3 : Use formula to calculate sample deviation Step 1 : Sum all the values Q1a. Calculate the mean : 592 ÷ 8 = 74 Q1a. Calculate the sample deviation

Nat 5 Created by Mr. Lafferty Maths Dept. Heart rate (x)x2x2 80 81 83 90 94 96 100 Totals 6400 6561 6889 8100 8836 9216 9216 10000 Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM 15-Apr-15 www.mathsrevision.com Standard Deviation For a Sample of Data ∑x = 720 Q1b(ii) Calculate the sample deviation Q1b(i) Calculate the mean : 720 ÷ 8 = 90 ∑x 2 = 65218

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Standard Deviation For a Sample of Data Q1b(iii) Who are fitter the athletes or staff. Compare means Athletes are fitter Staff Athletes Q1b(iv) What does the deviation tell us. Staff data is more spread out.

Nat 5 15-Apr-15Created by Mr. Lafferty Maths Dept. Now try N5 TJ Ex 11.5 Q3 onwards Ch11 (page 113) www.mathsrevision.com Standard Deviation For a FULL set of Data

Calculate the mean and standard deviation

Go on to next slide for part c

Qs b next slide

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