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**Created by Mr. Lafferty Maths Dept.**

Statistics S5 Int2 Quartiles from a Frequency Table Quartiles from a Cumulative Frequency Table Estimating Quartiles from C.F Graphs Standard Deviation Standard Deviation from a sample Scatter Graphs Probability Relative Frequency & Probability 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr Lafferty Maths Dept**

Starter Questions S5 Int2 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr Lafferty Maths Dept**

Statistics Quartiles from Frequency Tables S5 Int2 Learning Intention Success Criteria To explain how to calculate quartiles from frequency tables. Know the term quartiles. Calculate quartiles given a frequency table. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr Lafferty Maths Dept**

Statistics Quartiles from Frequency Tables S5 Int2 Reminder ! 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. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr Lafferty Maths Dept**

Statistics Quartiles from Frequency Tables 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 = 4 R0 6 number Q1 6 number Q2 6 number Q3 6 number The quartiles fall in the gaps between Q1 : the 6th and 7th numbers Q2 : the 12th and 13th numbers Q3 : the 18th and 19th numbers. 31-Mar-17 Created by Mr Lafferty Maths Dept

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Statistics Quartiles from Frequency Tables S5 Int2 Example 2 : For a list of 25 numbers, 25 ÷ 4 = 6 R1 6 number Q1 6 number 1 No. 6 number Q3 6 number Q2 The quartiles fall in the gaps between Q1 : the 6th and 7th Q2 : the 13th Q3 : the 19th and 20th numbers. 31-Mar-17 Created by Mr Lafferty Maths Dept

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Statistics Quartiles from Frequency Tables S5 Int2 Example 3 : For a list of 26 numbers, 26 ÷ 4 = 6 R2 6 number 1 No. 6 number Q2 6 number 1 No. 6 number Q1 Q3 The quartiles fall in the gaps between Q1 : the 7th number Q2 : the 13th and 14th number Q3 : the 20th number. 31-Mar-17 Created by Mr Lafferty Maths Dept

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Statistics Quartiles from Frequency Tables S5 Int2 Example 4 : For a list of 27 numbers, 27 ÷ 4 = 6 R3 6 number 1 No. 6 number 1 No. 6 number 1 No. 6 number Q1 Q2 Q3 The quartiles fall in the gaps between Q1 : the 7th number Q2 : the 14th number Q3 : the 21th number. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr Lafferty Maths Dept**

Statistics Quartiles from Frequency Tables S5 Int2 Example 4 : For a ordered list of 34. Describe the quartiles. 34 ÷ 4 = 8 R2 8 number 1 No. 8 number Q2 8 number 1 No. 8 number Q1 Q3 The quartiles fall in the gaps between Q1 : the 9th number Q2 : the 17th and 18th number Q3 : the 26th number. 31-Mar-17 Created by Mr Lafferty Maths Dept

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Statistics Quartiles from Frequency Tables S5 Int2 Now try Exercise 1 Start at 1b Ch11 (page 162) 31-Mar-17 Created by Mr Lafferty Maths Dept

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Starter Questions S5 Int2 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Statistics S5 Int2 Quartiles from Cumulative Frequency Table Learning Intention Success Criteria 1. To explain how to calculate quartiles from Cumulative Frequency Table. Find the quartile values from Cumulative Frequency Table. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Statistics S5 Int2 Quartiles from Cumulative Frequency Table Example 1 : The frequency table shows the length of phone calls ( in minutes) made from an office in one day. Time Freq. (f) Cum. Freq. 1 2 2 2 3 5 3 5 10 4 8 18 5 4 22 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Statistics S5 Int2 Quartiles from Cumulative Frequency Table 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 number 1 No. 5 number Q2 5 number 1 No. 5 number Q1 Q3 The quartiles fall in the gaps between Q1 : the 6th number Q1 : 3 minutes Q2 : the 11th and 12th number Q2 : 4 minutes Q3 : the 17th number. Q3 : 4 minutes 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Statistics S5 Int2 Quartiles from Cumulative Frequency Table Example 2 : A selection of schools were asked how many 5th year sections they have. Opposite is a table of the results. Calculate the quartiles for the results. No. Of Sections Freq. (f) Cum. Freq. 4 3 3 5 5 8 6 8 16 7 9 25 8 8 33 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Statistics S5 Int2 Quartiles from Cumulative Frequency Table We use a combination of quartiles from a frequency table and the Cumulative Frequency Column. Example 2 : For a list of 33 numbers, 33 ÷ 4 = 8 R1 8 number Q1 8 number 1 No. 8 number Q3 8 number Q2 The quartiles fall in the gaps between Q1 : the 8th and 9th numbers Q1 : 5.5 Q2 : the 17th number Q2 : 7 Q3 : the 25th ad 26th numbers. Q3 : 7.5 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Statistics S5 Int2 Quartiles from Cumulative Frequency Table Now try Exercise 2 Ch11 (page 163) 31-Mar-17 Created by Mr Lafferty Maths Dept

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Starter Questions S5 Int2 2cm 3cm 29o 4cm A 70o C 53o 8cm B 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Quartiles from Cumulative Frequency Graphs**

S5 Int2 Learning Intention Success Criteria 1. To show how to estimate quartiles from cumulative frequency graphs. Know the terms quartiles. 2. Estimate quartiles from cumulative frequency graphs. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Quartiles from Cumulative Frequency Graphs**

S5 Int2 Number of sockets Cumulative Frequency 10 2 20 9 30 24 40 34 50 39 60

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**Cumulative Frequency Graphs**

New Term Interquartile range Semi-interquartile range (Q3 – Q1 )÷2 = ( )÷2 =7.5 Cumulative Frequency Graphs S5 Int2 Quartiles 40 ÷ 4 =10 Q3 Q3 =36 Q2 Q2 =27 Q1 Q1 =21

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**Quartiles from Cumulative Frequency Graphs**

S5 Int2 Km travelled on 1 gallon (mpg) Cumulative Frequency 20 3 25 11 30 35 53 40 69 45 76 50 80

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**Cumulative Frequency Graphs Cumulative Frequency Graphs**

New Term Interquartile range Semi-interquartile range (Q3 – Q1 )÷2 = ( )÷2 =4.5 Cumulative Frequency Graphs Cumulative Frequency Graphs S5 Int2 Q3 = 37 Quartiles 80 ÷ 4 =20 Q2 = 32 Q1 =28

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**Quartiles from Cumulative Frequency Graphs**

S5 Int2 Now try Exercise 3 Ch11 (page 166) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Starter Questions S5 Int2 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Standard Deviation S5 Int2 Learning Intention Success Criteria 1. To explain the term and calculate the Standard Deviation for a collection of data. Know the term Standard Deviation. Calculate the Standard Deviation for a collection of data. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Standard Deviation For a FULL set of Data**

S5 Int2 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. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Standard Deviation For a FULL set of Data S5 Int2 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. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Standard Deviation For a FULL set of Data**

Step 1 : Find the mean 375 ÷ 5 = 75 Step 5 : Take the square root of step 4 √13.6 = 3.7 Standard Deviation is 3.7 (to 1d.p.) Step 2 : Score - Mean Step 4 : Mean square deviation 68 ÷ 5 = 13.6 Standard Deviation For a FULL set of Data Step 3 : (Deviation)2 S5 Int2 Example 1 : Find the standard deviation of these five scores 70, 72, 75, 78, 80. Score Deviation (Deviation)2 70 72 75 78 80 Totals 375 -5 25 -3 9 3 9 5 25 68 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Standard Deviation For a FULL set of Data**

Step 1 : Find the mean 180 ÷ 6 = 30 Step 5 : Take the square root of step 4 √ = 12.7 (to 1d.p.) Standard Deviation is £12.70 Step 2 : Score - Mean Step 4 : Mean square deviation 962 ÷ 6 = Step 3 : (Deviation)2 Standard Deviation For a FULL set of Data S5 Int2 Example 2 : Find the standard deviation of these six amounts of money £12, £18, £27, £36, £37, £50. Score Deviation (Deviation)2 12 18 27 36 37 50 Totals 180 -18 324 -12 144 -3 9 6 36 7 49 20 400 962 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Standard Deviation For a FULL set of Data**

S5 Int2 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. Mean Mean 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Standard Deviation S5 Int2 Now try Exercise 4 Ch11 (page 169) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Starter Questions S5 Int2 Waist Sizes Frequency 28” 7 30” 12 32” 23 34” 14 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Standard Deviation For a Sample of Data S5 Int2 Learning Intention Success Criteria 1. To show how to calculate the Standard deviation for a sample of data. Construct a table to calculate the Standard Deviation for a sample of data. 2. Use the table of values to calculate Standard Deviation of a sample of data. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Standard Deviation For a Sample of Data**

We will use this version because it is easier to use in practice ! S5 Int2 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 ∑ = The sum of x = sample mean n = number in sample 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Standard Deviation For a Sample of Data**

Q1a. Calculate the mean : 592 ÷ 8 = 74 Step 1 : Sum all the values Step 3 : Use formula to calculate sample deviation Step 2 : Square all the values and find the total Q1a. Calculate the sample deviation Standard Deviation For a Sample of Data S5 Int2 Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76. Heart rate (x) x2 70 72 73 74 75 76 Totals 4900 5184 5329 5476 5625 5776 5776 5776 31-Mar-17 Created by Mr. Lafferty Maths Dept. ∑x = 592 ∑x2 = 43842

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**Standard Deviation For a Sample of Data**

Q1b(i) Calculate the mean : 720 ÷ 8 = 90 Q1b(ii) Calculate the sample deviation Standard Deviation For a Sample of Data S5 Int2 Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM Heart rate (x) x2 80 81 83 90 94 96 100 Totals 6400 6561 6889 8100 8836 9216 9216 10000 31-Mar-17 Created by Mr. Lafferty Maths Dept. ∑x = 720 ∑x2 = 65218

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**Standard Deviation For a Sample of Data**

Q1b(iii) Who are fitter the athletes or staff. Compare means Athletes are fitter Q1b(iv) What does the deviation tell us. Staff data is more spread out. Standard Deviation For a Sample of Data S5 Int2 Athletes Staff 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Standard Deviation For a Sample of Data S5 Int2 Now try Ex 5 & 6 Ch11 (page 171) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Starter Questions S5 Int2 33o 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Scatter Graphs S5 Int2 Construction of Scatter Graphs Learning Intention Success Criteria Construct and understand the Key-Points of a scattergraph. To construct and interpret Scattergraphs. 2. Know the term positive and negative correlation. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Scatter Graphs www.mathsrevision.com Construction of Scatter Graph**

This scattergraph shows the heights and weights of a sevens football team Scatter Graphs Write down height and weight of each player. S5 Int2 Construction of Scatter Graph Bob Tim Joe Sam Gary Dave Jim 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr Lafferty Maths Dept**

Scatter Graphs S5 Int2 Construction of Scatter Graph When two quantities are strongly connected we say there is a strong correlation between them. Best fit line x x Best fit line Strong positive correlation Strong negative correlation 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Scatter Graphs www.mathsrevision.com Construction of Scatter Graph**

S5 Int2 Construction of Scatter Graph Key steps to: Drawing the best fitting straight line to a scatter graph Plot scatter graph. 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. 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Scatter Graphs www.mathsrevision.com Construction of Scatter Graph**

Find the mean for theAge and Prices values. Draw in the best fit line Mean Age = 2.9 Mean Price = £6000 Scatter Graphs S5 Int2 Construction of Scatter Graph Is there a correlation? If yes, what kind? Age Price (£1000) 3 1 2 4 5 9 8 7 6 Strong negative correlation 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Scatter Graphs www.mathsrevision.com Construction of Scatter Graph**

S5 Int2 Construction of Scatter Graph Key steps to: Finding the equation of the straight line. Pick any 2 points of graph ( pick easy ones to work with). Calculate the gradient using : Find were the line crosses y–axis this is b. Write down equation in the form : y = ax + b 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Crosses y-axis at 10 Scatter Graphs S5 Int2 Pick points (0,10) and (3,6) y = -1.33x + 10 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr. Lafferty Maths Dept.**

Scatter Graphs S5 Int2 Construction of Scatter Graph Now try Exercise 7 Ch11 (page 175) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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Starter Questions S5 Int2 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr Lafferty Maths Dept**

Probability S5 Int2 Learning Intention Success Criteria To understand probability in terms of the number line and calculate simple probabilities. Understand the probability line. Calculate simply probabilities. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability Likelihood Line**

S5 Int2 1 0.5 Impossible Evens Certain Not very likely Very likely Seeing a butterfly In July School Holidays Winning the Lottery Baby Born A Boy Go back in time 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability Likelihood Line**

S5 Int2 1 0.5 Impossible Evens Certain Not very likely Very likely It will Snow in winter Homework Every week Everyone getting 100 % in test Toss a coin That land Heads Going without Food for a year. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability We can normally attach a value**

S5 Int2 We can normally attach a value to the probability of an event happening. To work out a probability P(A) = Probability is ALWAYS in the range 0 to 1 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability Number Likelihood Line**

S5 Int2 1 2 3 5 4 7 6 8 1 0.5 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 Impossible Evens Certain 8 P = Q. What is the chance of picking a number between 1 – 8 ? = 1 8 4 Q. What is the chance of picking a number that is even ? P(E) = = 0.5 8 Q. What is the chance of picking the number 1 ? 1 P(1) = = 0.125 31-Mar-17 Created by Mr Lafferty Maths Dept 8

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**Probability Likelihood Line**

S5 Int2 52 cards in a pack of cards 1 0.5 0.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 Impossible Evens Certain Not very likely Very likely 26 Q. What is the chance of picking a red card ? P (Red) = = 0.5 52 13 Q. What is the chance of picking a diamond ? P (D) = = 0.25 52 4 Q. What is the chance of picking ace ? P (Ace) = = 0.08 52 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr. Lafferty Maths Dept.**

Probability S5 Int2 Now try Ex 8 Ch11 (page 177) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr Lafferty Maths Dept**

Starter Questions S5 Int2 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr Lafferty Maths Dept**

Relative Frequencies S5 Int2 Learning Intention Success Criteria To understand the term relative frequency. Know the term relative frequency. Calculate relative frequency from data given. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Relative Frequencies Relative Frequency www.mathsrevision.com**

Relative Frequency always added up to 1 S5 Int2 Relative Frequency How often an event happens compared to the total number of events. Example : Wine sold in a shop over one week Country Frequency Relative Frequency France 180 Italy 90 Spain Total 180 ÷ 360 = 0.5 90 ÷ 360 = 0.25 90 ÷ 360 = 0.25 360 1 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Relative Frequencies www.mathsrevision.com Example**

S5 Int2 Example Calculate the relative frequency for boys and girls born in the Royal Infirmary hospital in December 2007. Boys Girls Total Frequency 300 200 Relative Frequency Relative Frequency adds up to 1 500 0.6 0.4 1 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Created by Mr. Lafferty Maths Dept.**

Relative Frequencies S5 Int2 Now try Ex 9 Ch11 (page 179) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Created by Mr. Lafferty Maths Dept.**

Starter Questions S5 Int2 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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**Probability from Relative Frequency**

S5 Int2 Learning Intention Success Criteria To understand the connection of probability and relative frequency. Know the term relative frequency. Estimate probability from the relative frequency. 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability from Relative Frequency**

When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome Probability from Relative Frequency S5 Int2 Example 1 Three students carry out a survey to study left handedness in a school. Results are given below Number of Left - Hand Students Total Asked Relative Frequency Sean 2 10 Karen 3 25 Daniel 20 200 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability from Relative Frequency**

Who’s results would you use as a estimate of the probability of a house being alarmed ? Megan’s Probability from Relative Frequency S5 Int2 Example 2 Three students carry out a survey to study how many houses had an alarm system in a particular area. Results are given below What is the probability that a house is alarmed ? 0.4 Number of Alarmed Houses Total Asked Relative Frequency Paul 7 10 Amy 12 20 Megan 40 100 31-Mar-17 Created by Mr Lafferty Maths Dept

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**Probability from Relative Frequency**

S5 Int2 Now try Ex 10 Ch11 Start at Q2 (page 181) 31-Mar-17 Created by Mr. Lafferty Maths Dept.

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