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Statistics S3 Credit www.mathsrevision.com Mean Mean from a Frequency Table Median and Mode Range of a Set of Data Semi-Interquartile Range ( SIQR ) Quartile Graphs ( S – Curves ) Standard Deviation / Sample Standard Deviation Probability Estimating Probability from Relative Frequency 6-May-17 Created by Mr. Lafferty 1 Starter Questions www.mathsrevision.com S3 Credit Q1. Round to 2 significant figures (a) 52.567 (b) 626 Q2. Why is 2 + 4 x 2 = 10 and not 12 Q3. Solve for x 2x 20 8 x 6-May-17 Created by Mr. Lafferty 2 Frequency Tables www.mathsrevision.com S3 Credit Working Out the Mean Learning Intention 1. Explain the meaning of the term Mean Success Criteria 1. Know the term mean. 2. Calculate the mean for a given set of data. 6-May-17 Created 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. Mean = Sum of values Number of values For example, the mean of 3, 6, 7, 9 and 9 is 6-May-17 3+6+7+9+9 34 5 5 6.8 Created by Mr. Lafferty 4 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 7 + 5 + 2 + 7 + 6 + 12 + 10 + 4 + 8 + 9 10 70 10 6-May-17 7 Created by Mr. Lafferty 5 Average / Mean www.mathsrevision.com S3 Credit Now try Exercise 2.1 Ch12 (page 228) 6-May-17 Created by Mr. Lafferty 6 Starter Questions S3 Credit www.mathsrevision.com 1. Mutliply out 2y( y - 4) 2. Explain why 9 x 2 36 factorises to 9( x - 2)( x 2) 3. 12% of £22 4. Tidy up the expression 7 - (-10) 3 6-May-17 Created by Mr. Lafferty Maths Dept. Frequency Tables www.mathsrevision.com S3 Credit Working Out the Mean Learning Intention 1. To explain how to work out the Mean by adding in a third column to a Frequency Table. Success Criteria 1. Add a third column to a frequency table. 2. Calculate the Mean from a frequency Table. 6-May-17 Created by Mr. Lafferty Maths Dept. Frequency Tables www.mathsrevision.com S3 Credit Working Out the Mean Example : This table shows the number of light bulbs used in people’s living rooms Adding a third column to this table will help us find the total number of bulbs and the ‘Mean’. No of Bulbs (c) Freq. (f) 1 7 7x1=7 2 5 5 x 2 = 10 3 5 5 x 3 = 15 4 2 2x4=8 5 1 1x5=5 Totals 20 Mean Number of bulbs 45 = 2.25 bulbs per room 20 6-May-17 Created by Mr. Lafferty Maths Dept. (f) x (B) 45 Frequency Tables www.mathsrevision.com S3 Credit Working Out the Mean Example : This table shows the number of brothers and sisters of pupils in an S3 class. No of Sibling s (S) Adding a third column to this table will help us find the total number of siblings and the ‘Mean’. Mean Number of siblings = 6-May-17 33 = 1.1 siblings 30 Sxf 0 9 1 13 1 x 13 = 13 2 6 2 x 6 = 12 3 1 3x1=3 5 1 5x1=5 Totals Created by Mr. Lafferty Maths Dept. Freq. (f) 30 0 x 9 =0 33 Frequency Tables www.mathsrevision.com S3 Credit Working Out the Mean Now try Ex 2.2 Ch12 (page 229) 6-May-17 Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com S3 Credit Q1. 1.5% of £500 Q2. Find the ratio of cos 60o Q3. 75.9 x 70 Q4. Explain why the length a = 36m 6-May-17 Created by Mr. Lafferty 30m 24m a 12 Different Averages www.mathsrevision.com S3 Credit Learning Intention 1. To define the terms Median and Mode for a set of data. 6-May-17 Success Criteria 1. Know the terms Median and Mode. 2. Work out values for the Median and Mode for given set of data Created by Mr. Lafferty Maths Dept. Statistics S3 Credit 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. 6-May-17 Created by Mr Lafferty Maths Dept Different Averages S3 Credit www.mathsrevision.com Example : Find the median and mode for the set of data. 10, 2, 14, 1, 14, 7 Median = 1,2, 7,10,14,14 7 + 10 17 Median = = = 8.5 2 2 6-May-17 Created by Mr. Lafferty Maths Dept. Mode = 14 Different Averages www.mathsrevision.com S3 Credit Now try Exercise 3.1 Ch12 (page 231) 6-May-17 Created by Mr. Lafferty Maths Dept. Lesson Starter www.mathsrevision.com S3 Credit Q1. Explain why 2.5% of £800 = £20 Q2. Calculate sin 90o Q3. Factorise 5y2 – 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. 6-May-17 Created by Mr. Lafferty 17 Different Averages www.mathsrevision.com S3 Credit Learning Intention 1. To define the term Range for a set of data. Success Criteria 1. Know the term Range. 2. Calculate the value for the Range for given set of data 6-May-17 Created 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 small; the values are similar in size. When the range is large; the values vary widely in size. 6-May-17 Created by Mr. Lafferty 19 The Range www.mathsrevision.com S3 Credit Example : find the range for the following (a) 3, 1, 4, 10 10 – 1 = 9 (b) -3, 8, -6, 1, 7, 5 7 – (-6) = 13 – (-15.5) (c) The highest and lowest every recorded35.3 temperature = 50.8oC for Glasgow are 35.3oC and -15.5oC respectively. Find the value of the range. 6-May-17 Created by Mr. Lafferty 20 Statistics www.mathsrevision.com S3 Credit Working Out Statistics Now try Ex 4.1 Ch12 (page 232) 6-May-17 Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com S3 Credit 1. Calculate the mean, median, mode and range for the weekly wages £200, £100, £800 £160, £100, £380, £120 and £180. 2. Find the angle cos-1 (0). 3. Show that y2 5 y 6 factorises to ( y - 6)( y 1) 6-May-17 Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S3 Credit Semi- Inter Quartile Range Learning Intention 1. To explain the term semi-interquartile range. Success Criteria 1. Know the term semiinterquartile range. 2. Calculate semi-interquartile range. ( Q3 – Q1 ) ÷ 2 6-May-17 Created by Mr Lafferty Maths Dept Statistics S3 Credit Semi- Inter Quartile Range 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. Q1 = 25% Q2 = 50% Q3 = 75% 6-May-17 Created by Mr Lafferty Maths Dept Statistics www.mathsrevision.com S3 Credit Semi-interquartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2 = ( 9 – 3 ) ÷ 2Semi- Inter Quartile Range =3 Example 2 : For a list of 9 numbers find the SIQR R1 3, 3, 7, 8, 10, 9, 1, 5, 9 9 ÷ 4 = 2 1 3 2 number Q1 3 5 7 2 number 1 No. Q2 8 9 9 2 number 2 number Q3 The quartiles fall in the gaps between Q1 : the 3rd and 4th numbers 3 Q2 : the 5th number 7 Q3 : the 7th and 8th number. 9 6-May-17 Created by Mr Lafferty Maths Dept 10 Statistics Semi-interquartile Range SemiInter Quartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2 = ( 10 – 3 ) ÷ 2 = 3.5 www.mathsrevision.com S3 Credit Example 3 : For the ordered list find the SIQR. R3 3, 6, 2, 10, 12, 3, 4 7 ÷ 4 = 1 2 3 1 number 4 1 number Q1 6-May-17 3 6 10 1 number Q2 1 number Q3 The quartiles fall in the gaps between Q1 : the 2th number 3 Q2 : the 4th number 4 Q3 : the 6th number. 10 Created by Mr Lafferty Maths Dept 12 Statistics www.mathsrevision.com S3 Credit Semi- Inter Quartile Range Now try Ex 5.1 Ch12 (page 235) 6-May-17 Created by Mr Lafferty Maths Dept Starter Questions S3 Credit www.mathsrevision.com 1. Is the following statment true? 4(y + 3) - 3(8 - x) = 4y -12 + 3x 2. Find the angle for sin -1 (0.7) 3. Explain why I can simply pick out the quartiles from this dataset. 10, 12, 14, 18, 22, 30,32 6-May-17 Created by Mr. Lafferty Maths Dept. Quartiles from Cumulative Frequency Graphs www.mathsrevision.com S3 Credit Learning Intention 1. To show how to estimate quartiles from cumulative frequency graphs. Success Criteria 1. Know the terms quartiles. 2. Estimate quartiles from cumulative frequency graphs. 6-May-17 Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S3 Credit Quartiles from Cumulative Frequency Graphs Number of sockets Frequency 10 20 30 40 50 60 2 7 15 10 5 1 Cumulative Frequency 2 9 24 34 39 40 Cumulative Frequency www.mathsrevision.com S3 Credit Interquartile Range = (36 - 21) = 15 Cumulative Frequency Semi-interquartile range SIQR = (QGraphs 3 – Q1 )÷2 = (36 - 21)÷2 =7.5 45 Quartiles 40 35 Q3 30 25 Q2 20 10 Q3 =36 Q2 =27 Q1 15 40 ÷ 4 =10 Q1 =21 5 0 0 10 20 30 40 50 Number of Sockets 60 70 www.mathsrevision.com S3 Credit Quartiles from Cumulative Frequency Graphs Km travelled on 1 gallon (kmpg) 20 25 30 35 40 45 50 Cumulative Frequency 3 11 30 53 69 76 80 Interquartile range = (37 - 28) = 9 Cumulative Frequency Semi-interquartile range = (Q 3 – Q1 ) ÷2 Graphs = (37 - 28) ÷2 = 4.5 Cumulative Frequency www.mathsrevision.com S3 Credit 90 80 Q3 = 37 70 60 Quartiles 80 ÷ 4 =20 Q2 = 32 50 40 Q1 =28 30 20 10 0 0 10 20 30 40 50 Km travelled on 1 gallon (mpg) 60 Statistics www.mathsrevision.com S3 Credit Working Out Statistics Now try Ex 5.2 Ch12 (page 238) 6-May-17 Created by Mr. Lafferty Maths Dept. Starter Questions S3 Credit www.mathsrevision.com 1. Construct a cumulative frequency table For the data below. Waist Sizes Frequency 28” 7 30” 12 32” 23 34” 14 2. A circle is divided into 6 equal pieces. Calculate the area of one of the pieces when the diameter of the circle is 20cm. 6-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation www.mathsrevision.com S3 Credit Learning Intention 1. To explain the term and calculate the Standard Deviation for a collection of data. 6-May-17 Success Criteria 1. Know the term Standard Deviation. 2. Calculate the Standard Deviation for a collection of data. Created by Mr. Lafferty Maths Dept. Standard Deviation For a FULL set of Data www.mathsrevision.com S3 Credit 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. 6-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation For a FULL set of Data S3 Credit www.mathsrevision.com 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. 6-May-17 Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S3 Credit Step25::Score - Mean Deviation Step Step 1 : FindStandard the mean Step 4 : Mean square deviation For a Take FULL set of Data the square root of step 4 375 ÷ 5 = 75 2 Step 3 : (Deviation)68 ÷ 5 = 13.6 √13.6 deviation = 3.7 Example 1 : Find the standard of these five scores 70, 72, 75, 78, 80. Standard Deviation is 3.7 (to 1d.p.) Score Deviation (Deviation)2 70 -5 25 72 -3 9 75 78 80 Totals 6-May-17 375 0 3 5 0 Created by Mr. Lafferty Maths Dept. 0 9 25 68 5Deviation : square deviation Step 1 : FindStandard the mean Step 4Step : Mean Step 2 : Score - Mean www.mathsrevision.com S3 Credit For a FULL set of Data 2 Take the square root of step 4 Step180 3 : ÷(Deviation) 6 = 30 962 ÷ 6 = 160.33 = 12.7 (to 1d.p.) Example 2 √160.33 : Find the standard deviation of these six amounts of money £12, £18, £27, £36, £37, £50. Standard Deviation is £12.70 Score Deviation (Deviation)2 12 -18 324 18 -12 144 27 36 37 Totals 6-May-17 -3 6 7 20 50 Created by Mr. Lafferty Maths Dept. 0 180 9 36 49 400 962 Standard Deviation For a FULL set of Data www.mathsrevision.com S3 Credit 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 6-May-17 Mean Created by Mr. Lafferty Maths Dept. Relative Frequencies www.mathsrevision.com S3 Credit Now try Ex 6.1 Ch12 (page 240) 6-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation For a Sample of Data www.mathsrevision.com S3 Credit Standard deviation Learning Intention 1. To show how to calculate the Sample Standard deviation for a sample of data. 6-May-17 Success Criteria 1. Know the term Sample Standard Deviation. 2. Calculate the Sample Standard Deviation for a collection of data. Created by Mr. Lafferty Maths Dept. www.mathsrevision.com S3 Credit Standard Deviation For a Sample ofWe Data will use this version because it is easier to use in a sample In real life situations it is normal to work with practice ). ! of data ( survey / questionnaire We can use two formulae to calculate the sample deviation. s ( x x) 2 n 1 s = standard deviation x = sample mean 6-May-17 x 2 s x n 1 ∑ = The sum of n = number in sample Created by Mr. Lafferty Maths Dept. n 2 www.mathsrevision.com 2: Q1a. Calculate the mean : Q1a.Step Calculate the Standard Deviation Step592 1 : ÷ 8 = 74 Step 3 :sample deviation Square all the values For a Sample of Data S3 find the total Credit Sum all the values Use formula toand calculate sample have deviation Example 1a : Eight athletes heart rates 70, 72, 73, 74, 75, 76, 76 and 76. s s 6-May-17 2 x x n 1 43842 Heart rate (x) 2 8 1 8 2 4900 72 43842 43808 5184 73 7 s n 592 70 x2 5329 74 5476 75 5625 76 s 4.875 5776 s76 2.2 (to 1 d . p5776 .) 76 Created by Mr. Lafferty Maths Dept. Totals ∑x = 592 5776 ∑x2 = 43842 www.mathsrevision.com S3 Credit Q1b(i) Calculate the mean : Standard Deviation Q1b(ii) Calculate the 720 ÷ 8 = 90 sample deviation For a Sample of Data Example 1b : Eight office staff train as athletes. Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM s s 6-May-17 x 2 x n 1 65218 2 s 81 90 2 94 96 96 7 418 s 7 s 7.7 100 Created by Mr. Lafferty Maths Dept. Totals ∑x = 720 x2 6400 65218 64800 83 720 8 1 80 n 8 Heart rate (x) 6561 6889 8100 8836 9216 9216 (to 1d10000 . p.) ∑x2 = 65218 Standard Q1b(iii) WhoDeviation are fitter Q1b(iv) What does the athletes or of staff. Forthe adeviation Sample Data tell us. www.mathsrevision.com S3 Credit Compare means Staff data is more spread Athletes are fitter out. Athletes Staff Mean 74 BPM Mean 90 BPM s 2.2 (to 1d. p.) s 7.7 (to 1d. p.) 6-May-17 Created by Mr. Lafferty Maths Dept. Standard Deviation For a Sample of Data www.mathsrevision.com S3 Credit Standard deviation Now try Ex 7.1 & 7.2 Ch12 (page 243) 6-May-17 Created by Mr. Lafferty Maths Dept. Starter Questions S3 Credit www.mathsrevision.com 1. If a triangle is right angled and two sides have length 10 and 9. What are the possible sizes of the third side. 2. Factorise x2 + 8x + 15 3. The missing angles are 90 and 57. Explain why? 6-May-17 Created by Mr. Lafferty Maths Dept. 33o Probability www.mathsrevision.com S3 Credit Learning Intention 1. To understand probability in terms of the number line and calculate simple probabilities. Success Criteria 1. Understand the probability line. 2. Calculate simply probabilities. 6-May-17 Created by Mr Lafferty Maths Dept Probability Likelihood Line www.mathsrevision.com S3 Credit 0 Impossible Seeing a butterfly In July 6-May-17 0.5 Not very likely School Holidays Evens Winning the Lottery Created by Mr Lafferty Maths Dept 1 Very likely Baby Born A Boy Certain Go back in time Probability Likelihood Line www.mathsrevision.com S3 Credit 0 Impossible It will Snow in winter 6-May-17 0.5 Not very likely Evens Homework Everyone getting Every week 100 % in test Created by Mr Lafferty Maths Dept 1 Very likely Certain Toss a coin Going without That land Food Heads for a year. Probability www.mathsrevision.com S3 Credit We can normally attach a value to the probability of an event happening. To work out a probability number of outcomes P(A) = Total number of possible outcomes Probability is ALWAYS in the range 0 to 1 6-May-17 Created by Mr Lafferty Maths Dept Probability Number Likelihood Line S3 Credit www.mathsrevision.com 1 0 2 0.1 Impossible 3 0.2 4 0.3 5 0.4 6 0.5 0.6 7 0.7 8 0.8 Evens P= Q. What is the chance of picking a number that is even ?P(E) = Q. What is the chance of picking the number 1 ? Created by Mr Lafferty Maths Dept 1 Certain Q. What is the chance of picking a number between 1 – 8 ? 6-May-17 0.9 P(1) = 8 =1 8 4 = 0.5 8 1 = 0.125 8 Probability Likelihood Line S3 Credit www.mathsrevision.com 52 cards in a pack of cards 0 0.1 Impossible 0.2 0.3 Not very likely 0.4 0.5 0.6 0.7 Evens 0.8 Very likely Q. What is the chance of picking a red card ? P (Red) = Q. What is the chance of picking a diamond ? P (D) = Q. What is the chance of picking ace ? P (Ace) = 4 52 6-May-17 Created by Mr Lafferty Maths Dept 0.9 1 Certain 26 52 = 0.5 13 = 0.25 52 = 0.08 Probability www.mathsrevision.com S3 Credit Now try Ex 8.1 Ch12 (page 246) 6-May-17 Created by Mr. Lafferty Maths Dept. Starter Questions www.mathsrevision.com S3 Credit 1. Is it true that 16x2 -36 factorises to 4(2x -3)(2x + 3) 2. Write down what you understand by the term (SOH)(C AH)(T OA) 6-May-17 Created by Mr Lafferty Maths Dept Probability from Relative Frequency www.mathsrevision.com S3 Credit Learning Intention 1. To understand the connection of probability and relative frequency. 6-May-17 Success Criteria 1. Know the term relative frequency. 2. Estimate probability from the relative frequency. Created by Mr Lafferty Maths Dept www.mathsrevision.com S3 Credit When the sum of the frequencies is LARGE Frequencies the Relative Relative relative frequency is a good Frequency estimate of the probability always added Relative Frequency of an outcome up to 1 How often an event happens compared to the total number of events. Example : Wine sold in a shop over one week Country Frequency France 180 Italy 90 90 ÷ 360 = 0.25 Spain 90 90 ÷ 360 = Total 360 1 6-May-17 Relative Frequency 180 ÷ 360 = 0.5 Created by Mr Lafferty Maths Dept 0.25 S3 Credit www.mathsrevision.com Example When the sum of the frequencies is LARGE the Relative Frequencies relative frequency is a good estimate of the probability of an outcome Calculate the relative frequency for boys and girls born in the Royal Infirmary hospital in December 2007. Boys Girls Total Frequency 300 200 500 Relative Frequency 0.6 0.4 1 6-May-17 Created by Mr Lafferty Maths Dept Relative Frequency adds up to 1 Relative Frequencies www.mathsrevision.com S3 Credit Now try Ex 8.2 Ch12 (page 248) 6-May-17 Created by Mr. Lafferty Maths Dept.