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Statistics

S5 Int2. Quartiles from a Frequency Table. Statistics. Quartiles from a Cumulative Frequency Table. Estimating Quartiles from C.F Graphs. Standard Deviation. Standard Deviation from a sample. www.mathsrevision.com. Scatter Graphs. Probability. Relative Frequency & Probability. S5 Int2.

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Statistics

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

  2. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr Lafferty Maths Dept

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

  4. Quartiles from Frequency Tables S5 Int2 Reminder ! Statistics 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. www.mathsrevision.com 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. Created by Mr Lafferty Maths Dept

  5. 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. Statistics Example 1 : For a list of 24 numbers, 24 ÷ 6 = 4 R0 6 number Q1 6 number Q2 6 number Q3 6 number www.mathsrevision.com 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. Created by Mr Lafferty Maths Dept

  6. Quartiles from Frequency Tables S5 Int2 Statistics Example 2 : For a list of 25 numbers, 25 ÷ 4 = 6 R1 1 No. 6 number Q1 6 number 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. www.mathsrevision.com Created by Mr Lafferty Maths Dept

  7. Quartiles from Frequency Tables S5 Int2 Statistics 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. www.mathsrevision.com Created by Mr Lafferty Maths Dept

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

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

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

  11. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

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

  14. 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 www.mathsrevision.com 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 Created by Mr. Lafferty Maths Dept.

  15. Statistics Quartiles from Cumulative Frequency Table S5 Int2 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. Cum. Freq. 4 3 3 5 5 8 6 8 16 www.mathsrevision.com 7 9 25 8 8 33 Created by Mr. Lafferty Maths Dept.

  16. 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 1 No. 8 number Q1 8 number 8 number Q3 8 number Q2 www.mathsrevision.com 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 Created by Mr. Lafferty Maths Dept.

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

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

  19. Quartiles fromCumulative FrequencyGraphs S5 Int2 Learning Intention Success Criteria • Know the terms quartiles. • 1. To show how to estimate quartiles from cumulative frequency graphs. • 2. Estimate quartiles from cumulative frequency graphs. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  20. Quartiles fromCumulative FrequencyGraphs S5 Int2 www.mathsrevision.com

  21. New Term Interquartile range Semi-interquartile range (Q3 – Q1 )÷2 = (36 - 21)÷2 =7.5 Cumulative FrequencyGraphs S5 Int2 Quartiles 40 ÷ 4 =10 Q3 Q3 =36 Q2 Q2 =27 www.mathsrevision.com Q1 Q1 =21

  22. Quartiles fromCumulative FrequencyGraphs S5 Int2 www.mathsrevision.com

  23. New Term Interquartile range Semi-interquartile range (Q3 – Q1 )÷2 = (37 - 28)÷2 =4.5 Cumulative FrequencyGraphs Cumulative FrequencyGraphs S5 Int2 Q3 = 37 Quartiles 80 ÷ 4 =20 Q2 = 32 www.mathsrevision.com Q1 =28

  24. Quartiles fromCumulative FrequencyGraphs S5 Int2 Now try Exercise 3 Ch11 (page 166) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  25. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

  27. 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. www.mathsrevision.com The semi-interquartile range is less sensitive to a single number changing but again it is only really based on two of the score. Created by Mr. Lafferty Maths Dept.

  28. 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. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  29. 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. -5 25 -3 9 www.mathsrevision.com 0 0 3 9 5 25 0 68 Created by Mr. Lafferty Maths Dept.

  30. Step 1 : Find the mean 180 ÷ 6 = 30 Step 5 : Take the square root of step 4 √160.33 = 12.7 (to 1d.p.) Standard Deviation is £12.70 Step 2 : Score - Mean Step 4 : Mean square deviation 962 ÷ 6 = 160.33 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. -18 324 -12 144 www.mathsrevision.com -3 9 6 36 7 49 20 400 962 0 Created by Mr. Lafferty Maths Dept.

  31. Standard Deviation For a FULL set of Data S5 Int2 When Standard Deviation is HIGH it means the data values are spread out from the MEAN. When Standard Deviation is LOW it means the data values are close to the MEAN. www.mathsrevision.com Mean Mean Created by Mr. Lafferty Maths Dept.

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

  33. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  34. Standard Deviation For a Sample of Data S5 Int2 Learning Intention Success Criteria • 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. • 2. Use the table of values to calculate Standard Deviation of a sample of data. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

  35. 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. www.mathsrevision.com s = standard deviation ∑ = The sum of x = sample mean n = number in sample Created by Mr. Lafferty Maths Dept.

  36. Q1a. Calculate the mean : 592 ÷ 8 = 74 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 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. 4900 5184 5329 www.mathsrevision.com 5476 5625 5776 5776 5776 Created by Mr. Lafferty Maths Dept. ∑x = 592 ∑x2 = 43842

  37. 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 6400 6561 6889 www.mathsrevision.com 8100 8836 9216 9216 10000 Created by Mr. Lafferty Maths Dept. ∑x = 720 ∑x2 = 65218

  38. 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 www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

  40. S5 Int2 Starter Questions www.mathsrevision.com 33o Created by Mr. Lafferty Maths Dept.

  41. 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. www.mathsrevision.com Created by Mr Lafferty Maths Dept

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

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

  44. Scatter Graphs Construction of Scatter Graph S5 Int2 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. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

  46. Scatter Graphs Construction of Scatter Graph S5 Int2 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 www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

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

  49. S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

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