1. 25-Jan-16 Quartiles from a Frequency Table Quartiles from a Cumulative Frequency Table Statistics Estimating Quartiles from C.F Graphs Standard Deviation Scatter Graphs Standard Deviation from a sample Probability Relative Frequency & Probability

2. 25-Jan-16 Starter Questions

3. 25-Jan-16 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. Quartiles from Frequency Tables

4. 25-Jan-16 Statistics 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.

5. Quartiles from Frequency Tables 25-Jan-16 Statistics 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.

6. Quartiles from Frequency Tables 25-Jan-16 Statistics 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.

7. Quartiles from Frequency Tables 25-Jan-16 Statistics 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.

8. Quartiles from Frequency Tables 25-Jan-16 Statistics 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.

9. Quartiles from Frequency Tables 25-Jan-16 Statistics 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

10. 25-Jan-16 Now try Exercise 1 Start at 1b Ch11 (page 162) Statistics Quartiles from Frequency Tables

11. 25-Jan-16 Starter Questions Starter Questions

12. 25-Jan-16 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.

13. 25-Jan-16TimeFreq.(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. Statistics Quartiles from Cumulative Frequency Table

14. 25-Jan-16 Statistics 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 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

15. 25-Jan-16 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. Statistics Quartiles from Cumulative Frequency Table

16. 25-Jan-16 Statistics Quartiles from Cumulative Frequency Table 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 Example 2 :For a list of 33 numbers, 33 ÷ 4 = 8 R1 8 number8 number8 number8 numberQ1 Q2 Q3 1 No.

17. 25-Jan-16 Now try Exercise 2 Ch11 (page 163) Statistics Quartiles from Cumulative Frequency Table

18. 25-Jan-16 Starter Questions Starter Questions A B C 8cm 53 o 70 o 4cm 2cm 3cm 29 o

19. 25-Jan-16 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

20. Number of sockets Cumulative Frequency 102 209 3024 4034 5039 6040

21. Q 3 Cumulative Frequency Graphs 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

22. Quartiles from Cumulative Frequency Graphs Km travelled on 1 gallon (mpg) Cumulative Frequency 203 2511 30 3553 4069 4576 5080

23. Cumulative Frequency Graphs Q 3 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

24. 25-Jan-16 Now try Exercise 3 Ch11 (page 166) Quartiles from Cumulative Frequency Graphs

25. 25-Jan-16 Starter Questions Starter Questions

26. 25-Jan-16 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 1.Calculate the Standard Deviation for a collection of data.

27. 25-Jan-16 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.

28. 25-Jan-16 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.

29. 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 25-Jan-16 -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.)

30. 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 25-Jan-16 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

31. 25-Jan-16 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

32. 25-Jan-16 Now try Exercise 4 Ch11 (page 169) Standard Deviation

33. 25-Jan-16 Starter Questions Starter Questions Waist SizesFrequency 28”7 30”12 32”23 34”14

34. 25-Jan-16 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. Standard Deviation For a Sample of Data 2.Use the table of values to calculate Standard Deviation of a sample of data.

35. 25-Jan-16 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 !

36. Example 1a : Eight athletes have heart rates 70, 72, 73, 74, 75, 76, 76 and 76. 25-Jan-16 Standard Deviation For a Sample of Data Heart rate (x) 70 72 73 74 75 76 -4 -2 -1 0 1 2 2 2 ∑(x-x 2 ) =34 ∑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 16 0 1 4 1 4 4

37. 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 25-Jan-16 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

38. 25-Jan-16 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.

39. 25-Jan-16 Now try Ex 5 & 6 Ch11 (page 171) Standard Deviation For a Sample of Data

40. 25-Jan-16 Starter Questions Starter Questions 33 o

41. 25-Jan-16 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.

42. 25-Jan-16Created by Mr Lafferty Maths Dept 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.

43. 25-Jan-16 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.

44. 25-Jan-16 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.

45. 25-Jan-16 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 Mean Age = 2.9 Mean Price = £6000 Find the mean for theAge and Prices values.

46. 25-Jan-16 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

47. 25-Jan-16 Scatter Graphs Pick points (0,10) and (3,6) y = 1.38x + 10 Crosses y-axis at 10

48. 25-Jan-16 Now try Exercise 7 Ch11 (page 175) Scatter Graphs Construction of Scatter Graph

49. 25-Jan-16 Starter Questions Starter Questions

50. 25-Jan-16 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.

51. Probability Likelihood Line 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 25-Jan-16

52. Probability Likelihood Line 10.50 CertainEvensImpossible Not very likely Very likely Everyone getting 100 % in test Regular Homework Toss a coin That land Heads It will Snow in winter Going without Food for a year. 25-Jan-16

53. Probability To work out a probability P(Event) = Probability is ALWAYS in the range 0 to 1 25-Jan-16 We can normally attach a value to the probability of an event happening.

54. Probability Number Likelihood Line 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(1-8) = P(E) = P(1) = 25-Jan-16 Eight numbered tokens in a bag

55. Probability Likelihood Line 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 an 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 25-Jan-16

56. Now try Ex 8 Ch11 (page 177) Probability

57. 25-Jan-16 Starter Questions

58. 25-Jan-16 Learning Intention Success Criteria 1.Know the term relative frequency. 1.To understand the term relative frequency. 2.Calculate relative frequency from data given. Relative Frequencies

59. 25-Jan-16 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

60. 25-Jan-16 Relative Frequencies BoysGirlsTotalFrequency300200 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

61. 25-Jan-16 Now try Ex 9 Ch11 (page 179) Relative Frequencies

62. 25-Jan-16 Starter Questions Starter Questions

63. 25-Jan-16 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.

64. Probability from Relative Frequency 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 25-Jan-16 When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

65. Probability from Relative Frequency Number of Alarmed Houses Total Asked RelativeFrequencyPaul710 Amy1220 Megan40100 Example 2 25-Jan-16 What is the probability that a house is alarmed ? 0.4 Who’s results would you use as a estimate of the probability of a house being alarmed ? Megan’s Three students carry out a survey to study how many houses had an alarm system in a particular area. Results are given below

66. 25-Jan-16 Now try Ex 10 Ch11 Start at Q2 (page 181) Probability from Relative Frequency