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29-May-14Created by Mr. Lafferty Maths Dept. Quartiles from a Frequency Table Quartiles from a Cumulative Frequency Table Statistics www.mathsrevision.com Estimating Quartiles from C.F Graphs Standard Deviation Scatter Graphs Standard Deviation from a sample Probability Relative Frequency & Probability S5 Int2

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29-May-14Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com S5 Int2

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29-May-14Created by Mr Lafferty Maths Dept Statistics Learning Intention Success Criteria 1.Know the term quartiles. 1.To explain how to calculate quartiles from frequency tables. 2.Calculate quartiles given a frequency table. www.mathsrevision.com Quartiles from Frequency Tables S5 Int2

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Quartiles from Frequency Tables 29-May-14Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Reminder ! S5 Int2 Range : The difference between highest and Lowest values. It is a measure of spread. Median :The middle value of a set of data. When they are two middle values the median is half way between them. Mode :The value that occurs the most in a set of data. Can be more than one value. Quartiles :The median splits into lists of equal length. The medians of these two lists are called quartiles.

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Quartiles from Frequency Tables 29-May-14Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S5 Int2 To find the quartiles of an ordered list you consider its length. You need to find three numbers which break the list into four smaller list of equal length. Example 1 :For a list of 24 numbers, 24 ÷ 6 = 4R0 6 number6 number6 number6 numberQ1Q2Q3 The quartiles fall in the gaps between Q 1 :the 6 th and 7 th numbers Q 2 :the 12 th and 13 th numbers Q 3 :the 18 th and 19 th numbers.

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Quartiles from Frequency Tables 29-May-14Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S5 Int2 Example 2 :For a list of 25 numbers, 25 ÷ 4 = 6 R1 6 number6 number6 number6 numberQ1 Q2 Q3 1 No. The quartiles fall in the gaps between Q 1 :the 6 th and 7 th Q 2 :the 13 th Q 3 :the 19 th and 20 th numbers.

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Quartiles from Frequency Tables 29-May-14Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S5 Int2 Example 3 :For a list of 26 numbers, 26 ÷ 4 = 6 R2 6 number6 number6 number6 number Q1 Q2 Q3 1 No. The quartiles fall in the gaps between Q 1 :the 7 th number Q 2 :the 13 th and 14 th number Q 3 :the 20 th number. 1 No.

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Quartiles from Frequency Tables 29-May-14Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S5 Int2 Example 4 :For a list of 27 numbers, 27 ÷ 4 = 6 R3 6 number6 number6 number6 number Q1 Q2 Q3 1 No. The quartiles fall in the gaps between Q 1 :the 7 th number Q 2 :the 14 th number Q 3 :the 21 th number. 1 No. 1 No.

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Quartiles from Frequency Tables 29-May-14Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S5 Int2 Example 4 :For a ordered list of 34. Describe the quartiles. 8 number8 number8 number8 number Q1 Q2 Q3 1 No. The quartiles fall in the gaps between Q 1 :the 9 th number Q 2 :the 17 th and 18 th number Q 3 :the 26 th number. 1 No. 34 ÷ 4 = 8R2

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29-May-14Created by Mr Lafferty Maths Dept Now try Exercise 1 Start at 1b Ch11 (page 162) www.mathsrevision.com Statistics S5 Int2 Quartiles from Frequency Tables

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria Statistics Quartiles from Cumulative Frequency Table 1. To explain how to calculate quartiles from Cumulative Frequency Table. 1.Find the quartile values from Cumulative Frequency Table. S5 Int2

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.comTimeFreq.(f) Example 1 : The frequency table shows the length of phone calls ( in minutes) made from an office in one day. 2 5 4 8 3 1 2 3 4 5 2 10 22 18 5 Cum. Freq. S5 Int2 Statistics Quartiles from Cumulative Frequency Table

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Statistics Quartiles from Cumulative Frequency Table S5 Int2 We use a combination of quartiles from a frequency table and the Cumulative Frequency Column. For a list of 22 numbers, 22 ÷ 4 = 5 R2 5 number5 number5 number5 number Q1 Q2 Q3 1 No. The quartiles fall in the gaps between Q 1 :the 6 th number Q 1 : 3 minutes 1 No. Q 2 :the 11 th and 12 th number Q 2 : 4 minutes Q 3 :the 17 th number. Q 3 : 4 minutes

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com No. Of Sections Freq.(f) Example 2 : A selection of schools were asked how many 5 th year sections they have. Opposite is a table of the results. Calculate the quartiles for the results. 3 8 8 9 5 4 5 6 7 8 3 16 33 25 8 Cum. Freq. S5 Int2 Statistics Quartiles from Cumulative Frequency Table

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Statistics Quartiles from Cumulative Frequency Table S5 Int2 We use a combination of quartiles from a frequency table and the Cumulative Frequency Column. The quartiles fall in the gaps between Q 1 :the 8 th and 9 th numbers Q 1 : 5.5 Q 2 :the 17 th number Q 2 : 7 Q 3 :the 25 th ad 26 th numbers. Q 3 : 7.5 Example 2 :For a list of 33 numbers, 33 ÷ 4 = 8 R1 8 number8 number8 number8 numberQ1 Q2 Q3 1 No.

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29-May-14Created by Mr Lafferty Maths Dept Now try Exercise 2 Ch11 (page 163) www.mathsrevision.com S5 Int2 Statistics Quartiles from Cumulative Frequency Table

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2 A B C 8cm 53 o 70 o 4cm 2cm 3cm 29 o

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1. To show how to estimate quartiles from cumulative frequency graphs. 1.Know the terms quartiles. 2.Estimate quartiles from cumulative frequency graphs. S5 Int2 Quartiles from Cumulative Frequency Graphs

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Number of sockets Cumulative Frequency 102 209 3024 4034 5039 6040 www.mathsrevision.com S5 Int2

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Q 3 Cumulative Frequency Graphs www.mathsrevision.com S5 Int2 Quartiles 40 ÷ 4 =10 Q 1 Q 2 Q 1 =21 Q 2 =27 Q 3 =36 New Term Interquartile range Semi-interquartile range (Q 3 – Q 1 )÷2 = (36 - 21)÷2 =7.5

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Quartiles from Cumulative Frequency Graphs Km travelled on 1 gallon (mpg) Cumulative Frequency 203 2511 30 3553 4069 4576 5080 www.mathsrevision.com S5 Int2

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Cumulative Frequency Graphs www.mathsrevision.com S5 Int2 Q 3 Cumulative Frequency Graphs Quartiles 80 ÷ 4 =20 Q 1 Q 2 =28 = 32 = 37 New Term Interquartile range Semi-interquartile range (Q 3 – Q 1 )÷2 = (37 - 28)÷2 =4.5

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Exercise 3 Ch11 (page 166) www.mathsrevision.com S5 Int2 Quartiles from Cumulative Frequency Graphs

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know the term Standard Deviation. 1. To explain the term and calculate the Standard Deviation for a collection of data. Standard Deviation S5 Int2 1.Calculate the Standard Deviation for a collection of data.

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 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.

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 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.

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ScoreDeviation(Deviation) 2 70 72 75 78 80 Totals375 Example 1 :Find the standard deviation of these five scores 70, 72, 75, 78, 80. S5 Int2 Standard Deviation For a FULL set of Data Step 1 : Find the mean 375 ÷ 5 = 75 Step 3 : (Deviation) 2 29-May-14Created 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.)

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Example 2 :Find the standard deviation of these six amounts of money £12, £18, £27, £36, £37, £50. S5 Int2 Standard Deviation For a FULL set of Data Step 1 : Find the mean 180 ÷ 6 = 30 29-May-14Created 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

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 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

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Exercise 4 Ch11 (page 169) www.mathsrevision.com S5 Int2 Standard Deviation

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2 Waist SizesFrequency 287 3012 3223 3414

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Construct a table to calculate the Standard Deviation for a sample of data. 1. To show how to calculate the Standard deviation for a sample of data. S5 Int2 Standard Deviation For a Sample of Data 2.Use the table of values to calculate Standard Deviation of a sample of data.

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 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 !

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Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76. 29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 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

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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 29-May-14 www.mathsrevision.com S5 Int2 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

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 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.

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Ex 5 & 6 Ch11 (page 171) www.mathsrevision.com S5 Int2 Standard Deviation For a Sample of Data

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2 33 o

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29-May-14Created by Mr Lafferty Maths Dept www.mathsrevision.com Learning Intention Success Criteria 1.To construct and interpret Scattergraphs. 1.Construct and understand the Key-Points of a scattergraph. Scatter Graphs Construction of Scatter Graphs 2. Know the term positive and negative correlation. S5 Int2

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29-May-14Created by Mr Lafferty Maths Dept www.mathsrevision.com Scatter Graphs Construction of Scatter Graph Sam Jim Tim Gary Joe Dave Bob This scattergraph shows the heights and weights of a sevens football team Write down height and weight of each player.

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29-May-14Created by Mr Lafferty Maths Dept www.mathsrevision.com Scatter Graphs Construction of Scatter Graph x x x x x x Strong positive correlation x x x x x x Strong negative correlation Best fit line Best fit line When two quantities are strongly connected we say there is a strong correlation between them. S5 Int2

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 Scatter Graphs Construction of Scatter Graph Key steps to: Drawing the best fitting straight line to a scatter graph 1.Plot scatter graph. 2.Calculate mean for each variable and plot the coordinates on the scatter graph. 3.Draw best fitting line, making sure it goes through mean values.

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29-May-14Created by Mr Lafferty Maths Dept www.mathsrevision.com Scatter Graphs Construction of Scatter Graph Is there a correlation? If yes, what kind? Age Price (£1000) 3 1 1 2 3 3 4 4 5 9 8 8 7 6 5 5 4 2 S t r o n g n e g a t i v e c o r r e l a t i o n Draw in the best fit line S5 Int2 Mean Age = 2.9 Mean Price = £6000 Find the mean for theAge and Prices values.

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29-May-14Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S5 Int2 Scatter Graphs Construction of Scatter Graph Key steps to: Finding the equation of the straight line. 1.Pick any 2 points of graph ( pick easy ones to work with). 2.Calculate the gradient using : 3.Find were the line crosses y–axis this is b. 4.Write down equation in the form : y = ax + b

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29-May-14Created by Mr Lafferty Maths Dept www.mathsrevision.com Scatter Graphs Pick points (0,10) and (3,6) y = -1.33x + 10 S5 Int2 Crosses y-axis at 10

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Exercise 7 Ch11 (page 175) www.mathsrevision.com S5 Int2 Scatter Graphs Construction of Scatter Graph

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2

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29-May-14Created by Mr Lafferty Maths Dept Probability Learning Intention Success Criteria 1.Understand the probability line. 1.To understand probability in terms of the number line and calculate simple probabilities. 2.Calculate simply probabilities. www.mathsrevision.com S5 Int2

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Probability Likelihood Line www.mathsrevision.com 10.50 CertainEvensImpossible Not very likely Very likely Winning the Lottery School Holidays Baby Born A Boy Seeing a butterfly In July Go back in time S5 Int2 29-May-14Created by Mr Lafferty Maths Dept

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Probability Likelihood Line www.mathsrevision.com 10.50 CertainEvensImpossible Not very likely Very likely Everyone getting 100 % in test Homework Every week Toss a coin That land Heads It will Snow in winter Going without Food for a year. S5 Int2 29-May-14Created by Mr Lafferty Maths Dept

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Probability www.mathsrevision.com To work out a probability P(A) = Probability is ALWAYS in the range 0 to 1 S5 Int2 29-May-14Created by Mr Lafferty Maths Dept We can normally attach a value to the probability of an event happening.

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Probability Number Likelihood Line www.mathsrevision.com 10.50 CertainEvensImpossible 12354 7 68 0.10.20.30.40.60.70.80.9 Q. What is the chance of picking a number between 1 – 8 ? Q. What is the chance of picking a number that is even ? Q. What is the chance of picking the number 1 ? 8 8 = 1 4 8 = 0.5 1 8 = 0.125 P = P(E) = P(1) = S5 Int2 29-May-14Created by Mr Lafferty Maths Dept

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Probability Likelihood Line www.mathsrevision.com 10.50 CertainEvensImpossible Not very likely Very likely Q. What is the chance of picking a red card ? Q. What is the chance of picking a diamond ? Q. What is the chance of picking ace ? 52 = 0.5 13 52 = 0.25 4 52 = 0.08 26 0.10.20.30.40.60.70.80.9 P (Red) = P (D) = P (Ace) = 52 cards in a pack of cards S5 Int2 29-May-14Created by Mr Lafferty Maths Dept

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Ex 8 Ch11 (page 177) www.mathsrevision.com S5 Int2 Probability

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29-May-14Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com S5 Int2

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29-May-14Created by Mr Lafferty Maths Dept Learning Intention Success Criteria 1.Know the term relative frequency. 1.To understand the term relative frequency. 2.Calculate relative frequency from data given. www.mathsrevision.com S5 Int2 Relative Frequencies

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29-May-14Created by Mr Lafferty Maths Dept www.mathsrevision.com Relative Frequency How often an event happens compared to the total number of events.CountryFrequency Relative Frequency France180 Italy90 Spain90 Total Example : Wine sold in a shop over one week 180 ÷ 360 = 90 ÷ 360 = 90 ÷ 360 = 360 0.25 0.25 0.5 1 Relative Frequency always added up to 1 S5 Int2 Relative Frequencies

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29-May-14Created by Mr Lafferty Maths Dept Relative Frequencies www.mathsrevision.comBoysGirlsTotalFrequency300200 Relative Frequency Example Calculate the relative frequency for boys and girls born in the Royal Infirmary hospital in December 2007. 500 0.6 0.4 1 Relative Frequency adds up to 1 S5 Int2

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Ex 9 Ch11 (page 179) www.mathsrevision.com S5 Int2 Relative Frequencies

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29-May-14Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S5 Int2

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29-May-14Created by Mr Lafferty Maths Dept Probability from Relative Frequency Learning Intention Success Criteria 1.Know the term relative frequency. 1.To understand the connection of probability and relative frequency. 2.Estimate probability from the relative frequency. www.mathsrevision.com S5 Int2

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Probability from Relative Frequency www.mathsrevision.com Three students carry out a survey to study left handedness in a school. Results are given below Number of Left - Hand Students Total Asked RelativeFrequencySean210 Karen325 Daniel20200 Example 1 S5 Int2 29-May-14Created by Mr Lafferty Maths Dept When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

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Probability from Relative Frequency www.mathsrevision.com Number of Alarmed Houses Total Asked RelativeFrequencyPaul710 Amy1220 Megan40100 Example 2 S5 Int2 29-May-14Created by Mr Lafferty Maths Dept What is the probability that a house is alarmed ? 0.4 Whos results would you use as a estimate of the probability of a house being alarmed ? Megans Three students carry out a survey to study how many houses had an alarm system in a particular area. Results are given below

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29-May-14Created by Mr. Lafferty Maths Dept. Now try Ex 10 Ch11 Start at Q2 (page 181) www.mathsrevision.com S5 Int2 Probability from Relative Frequency

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