# 3-May-15Created by Mr. Lafferty1 Statistics Mean Mean from a Frequency Table Range of a Set of Data www.mathsrevision.com Median and Mode Semi-Interquartile.

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3-May-15Created by Mr. Lafferty1 Statistics Mean Mean from a Frequency Table Range of a Set of Data www.mathsrevision.com Median and Mode Semi-Interquartile Range ( SIQR ) S3 Credit Quartile Graphs ( S – Curves ) Standard Deviation / Sample Standard Deviation Probability Estimating Probability from Relative Frequency

3-May-15Created by Mr. Lafferty2 www.mathsrevision.com Starter Questions Q1.Round to 2 significant figures Q2.Why is 2 + 4 x 2 = 10 and not 12 Q3.Solve for x (a)52.567(b)626 S3 Credit

3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Know the term mean. 1. Explain the meaning of the term Mean Frequency Tables Working Out the Mean 2.Calculate the mean for a given set of data. S3 Credit

3-May-15Created by Mr. Lafferty4 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

3-May-15Created by Mr. Lafferty5 Two dice were thrown 10 times and their scores were added together and recorded. Find the mean for this data. 7, 5, 2, 7, 6, 12, 10, 4, 8, 9 Mean

3-May-15Created by Mr. Lafferty6 Now try Exercise 2.1 Ch12 (page 228) www.mathsrevision.com Average / Mean S3 Credit

3-May-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S3 Credit

3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.Add a third column to a frequency table. 1. To explain how to work out the Mean by adding in a third column to a Frequency Table. Frequency Tables Working Out the Mean 2.Calculate the Mean from a frequency Table. S3 Credit

S3 Credit 3-May-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)

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com No of Sibling s ( S ) Freq.(f) Example : This table shows the number of brothers and sisters of pupils in an S3 class. 9 6 1 30 1 13 0 1 2 3 5 Totals Frequency Tables Working Out the Mean Adding a third column to this table will help us find the total number of siblings and the ‘Mean’. 0 x 9 =0 2 x 6 = 12 5 x 1 = 5 3 x 1 = 3 1 x 13 = 13 33 S x f

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 2.2 Ch12 (page 229) www.mathsrevision.com Frequency Tables Working Out the Mean

3-May-15Created by Mr. Lafferty12 www.mathsrevision.com Starter Questions Q1. Q2.Find the ratio of cos 60 o Q3.75.9 x 70 Q4. Explain why the length a = 36m 30m 24m a S3 Credit

3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.To define the terms Median and Mode for a set of data. 1.Know the terms Median and Mode. Different Averages S3 Credit 2.Work out values for the Median and Mode for given set of data

3-May-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Reminder ! 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. S3 Credit

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Different Averages Example : Find the median and mode for the set of data. 10, 2, 14, 1, 14, 7

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Exercise 3.1 Ch12 (page 231) www.mathsrevision.com Different Averages

3-May-15Created by Mr. Lafferty17 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. S3 Credit

3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1.To define the term Range for a set of data. 1.Know the term Range. Different Averages S3 Credit 2.Calculate the value for the Range for given set of data

3-May-15Created by Mr. Lafferty19 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.

3-May-15Created by Mr. Lafferty20 www.mathsrevision.com The Range Example : find the range for the following (a)3, 1, 4, 10 (b)-3, 8, -6, 1, 7, 5 (c)The highest and lowest every recorded temperature for Glasgow are 35.3 o C and -15.5 o C respectively. Find the value of the range. 10 – 1 = 9 7 – (-6) = 13 35.3 – (-15.5) = 50.8 o C S3 Credit

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 4.1 Ch12 (page 232) www.mathsrevision.com Statistics Working Out Statistics

3-May-15Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com S3 Credit

3-May-15Created by Mr Lafferty Maths Dept Statistics Learning Intention Success Criteria 1.Know the term semi- interquartile range. 1.To explain the term semi-interquartile range. 2.Calculate semi-interquartile range. ( Q 3 – Q 1 ) ÷ 2 www.mathsrevision.com S3 Credit Semi- Inter Quartile Range

3-May-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com 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 :Splits a dataset into 4 equal lengths. Q 1 = 25% Q 2 = 50% Q 3 = 75% S3 Credit Semi- Inter Quartile Range

3-May-15Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com Example 2 :For a list of 9 numbers find the SIQR 3, 3, 7, 8, 10, 9, 1, 5, 9 2 number2 number2 number 2 number Q1Q2Q3 The quartiles fall in the gaps between Q 1 :the 3 rd and 4 th numbers Q 2 :the 5 th number Q 3 :the 7 th and 8 th number. S3 Credit Semi- Inter Quartile Range Semi-interquartile Range (SIQR) = ( Q 3 – Q 1 ) ÷ 2 = ( 9 – 3 ) ÷ 2 = 3 1 No. 3 7 9 9 ÷ 4 = 2R1 1 3 3 5 7 8 9 9 10

3-May-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 fall in the gaps between Q 1 :the 2 th number Q 2 :the 4 th number Q 3 :the 6 th number. 7 ÷ 4 = 1R3 S3 Credit Semi- Inter Quartile Range 3 4 10 Semi-interquartile Range (SIQR) = ( Q 3 – Q 1 ) ÷ 2 = ( 10 – 3 ) ÷ 2 = 3.5 2 3 3 4 6 10 12

S3 Credit 3-May-15Created by Mr Lafferty Maths Dept Now try Ex 5.1 Ch12 (page 235) www.mathsrevision.com Statistics Semi- Inter Quartile Range

3-May-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com S3 Credit

3-May-15Created 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. Quartiles from Cumulative Frequency Graphs S3 Credit

Quartiles from Cumulative Frequency Graphs Number of sockets FrequencyCumulative Frequency 1022 2079 301524 401034 50539 60140 www.mathsrevision.com S3 Credit

Q 3 Cumulative Frequency Graphs www.mathsrevision.com Quartiles 40 ÷ 4 =10 Q 1 Q 2 Q 1 =21 Q 2 =27 Q 3 =36 S3 Credit Interquartile Range = (36 - 21) = 15 Semi-interquartile range SIQR = (Q 3 – Q 1 )÷2 = (36 - 21)÷2 =7.5

Quartiles from Cumulative Frequency Graphs Km travelled on 1 gallon (kmpg) Cumulative Frequency 203 2511 30 3553 4069 4576 5080 www.mathsrevision.com S3 Credit

Cumulative Frequency Graphs www.mathsrevision.com Q 3 Cumulative Frequency Graphs Quartiles 80 ÷ 4 =20 Q 1 Q 2 =28 = 32 = 37 Interquartile range = (37 - 28) = 9 Semi-interquartile range = (Q 3 – Q 1 ) ÷2 = (37 - 28) ÷2 = 4.5 S3 Credit

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 5.2 Ch12 (page 238) www.mathsrevision.com Statistics Working Out Statistics

3-May-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com Waist SizesFrequency 28”7 30”12 32”23 34”14 S3 Credit

3-May-15Created 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 2.Calculate the Standard Deviation for a collection of data. S3 Credit

S3 Credit 3-May-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.

S3 Credit 3-May-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.

S3 Credit 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 3-May-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.)

S3 Credit 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 3-May-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

S3 Credit 3-May-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

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 6.1 Ch12 (page 240) www.mathsrevision.com Relative Frequencies

3-May-15Created by Mr. Lafferty Maths Dept. www.mathsrevision.com Learning Intention Success Criteria 1. To show how to calculate the Sample Standard deviation for a sample of data. Standard Deviation For a Sample of Data Standard deviation S3 Credit 1.Know the term Sample Standard Deviation. 2.Calculate the Sample Standard Deviation for a collection of data.

S3 Credit 3-May-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 !

S3 Credit Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76. 3-May-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

S3 Credit 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 3-May-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

S3 Credit 3-May-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.

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 7.1 & 7.2 Ch12 (page 243) www.mathsrevision.com Standard Deviation For a Sample of Data Standard deviation

3-May-15Created by Mr. Lafferty Maths Dept. Starter Questions Starter Questions www.mathsrevision.com 33 o S3 Credit

3-May-15Created 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 S3 Credit

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 3-May-15Created by Mr Lafferty Maths Dept S3 Credit

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. 3-May-15Created by Mr Lafferty Maths Dept S3 Credit

Probability www.mathsrevision.com To work out a probability P(A) = Probability is ALWAYS in the range 0 to 1 3-May-15Created by Mr Lafferty Maths Dept We can normally attach a value to the probability of an event happening. S3 Credit

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) = 3-May-15Created by Mr Lafferty Maths Dept S3 Credit

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 3-May-15Created by Mr Lafferty Maths Dept S3 Credit

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 8.1 Ch12 (page 246) www.mathsrevision.com Probability

3-May-15Created by Mr Lafferty Maths Dept Starter Questions www.mathsrevision.com S3 Credit

3-May-15Created 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 S3 Credit

3-May-15Created 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 Relative Frequencies S3 Credit When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

3-May-15Created 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 S3 Credit When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

S3 Credit 3-May-15Created by Mr. Lafferty Maths Dept. Now try Ex 8.2 Ch12 (page 248) www.mathsrevision.com Relative Frequencies

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