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

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

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

4. Sum of values Mean= Number of values The mean Themeanis 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 Created by Mr. Lafferty

5. Two dice were thrown 10 times and their scores were added together and recorded. Find themeanfor this data. 7, 5, 2, 7, 6, 12, 10, 4, 8, 9 Mean Created by Mr. Lafferty

6. Average / Mean S3 Credit Now try Exercise 2.1 Ch12 (page 228) www.mathsrevision.com Created by Mr. Lafferty

7. S3 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

8. Frequency Tables S3 Credit Working Out the Mean Learning Intention Success Criteria • 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. • 2. Calculate the Mean from a frequency Table. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

9. Frequency Tables Working Out the Mean Example : This table shows the number of light bulbs used in people’s living rooms (f) x (B) Adding a third column to this table will help us find the total number of bulbs and the ‘Mean’. 1 7 7 x 1 = 7 2 5 5 x 2 = 10 3 5 5 x 3 = 15 www.mathsrevision.com 4 2 2 x 4 = 8 5 1 1 x 5 = 5 Totals 20 45 Created by Mr. Lafferty Maths Dept.

10. Frequency Tables Working Out the Mean Example : This table shows the number of brothers and sisters of pupils in an S3 class. S x f Adding a third column to this table will help us find the total number of siblings and the ‘Mean’. 0 9 0 x 9 =0 1 13 1 x 13 = 13 2 6 2 x 6 = 12 www.mathsrevision.com 3 1 3 x 1 = 3 5 1 5 x 1 = 5 33 Totals 30 Created by Mr. Lafferty Maths Dept.

11. Frequency Tables Working Out the Mean Now try Ex 2.2 Ch12 (page 229) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

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

14. S3 Credit Reminder ! Statistics 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. Created by Mr Lafferty Maths Dept

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

16. Different Averages Now try Exercise 3.1 Ch12 (page 231) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

17. Lesson Starter S3 Credit Q1. Q2. Calculate sin 90o www.mathsrevision.com Q3. Factorise 5y2 – 10y A circle is divided into 10 equal pieces. Find the arc length of one piece of the circle if the radius is 5cm. Q4. Created by Mr. Lafferty

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

19. Range = highest value – lowest value Finding therange Therangeof 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. When the range is small; the values are similar in size. When the range is large; the values vary widely in size. Created by Mr. Lafferty

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

21. Statistics Working Out Statistics Now try Ex 4.1 Ch12 (page 232) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

22. S3 Credit Starter Questions www.mathsrevision.com Created by Mr Lafferty Maths Dept

23. S3 Credit Semi- Inter Quartile Range Statistics Learning Intention Success Criteria • Know the term semi-interquartile range. • To explain the term • semi-interquartile range. • Calculate • semi-interquartile range. • ( Q3 – Q1 ) ÷ 2 www.mathsrevision.com Created by Mr Lafferty Maths Dept

24. S3 Credit Semi- Inter Quartile Range 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 : Splits a dataset into 4 equal lengths. Q1 = 25% Q2 = 50% Q3 = 75% Created by Mr Lafferty Maths Dept

25. Semi-interquartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2 = ( 9– 3) ÷ 2 = 3 S3 Credit Semi- Inter Quartile Range Statistics Example 2 : For a list of 9 numbers find the SIQR 3, 3, 7, 8, 10, 9, 1, 5, 9 9 ÷ 4 = 2 R1 1 3 3 5 7 8 9 9 10 2 number 2 number 2 number 1 No. 2 number Q1 Q2 Q3 The quartiles fall in the gaps between Q1 : the 3rd and 4th numbers Q2 : the 5th number Q3 : the 7th and 8th number. www.mathsrevision.com 3 7 9 Created by Mr Lafferty Maths Dept

26. Semi-interquartile Range (SIQR) = ( Q3 – Q1 ) ÷ 2 = ( 10 – 3 ) ÷ 2 = 3.5 S3 Credit Semi- Inter Quartile Range Statistics Example 3 : For the ordered list find the SIQR. 3, 6, 2, 10, 12, 3, 4 7 ÷ 4 = 1 R3 2 3 3 4 6 10 12 1 number 1 number 1 number 1 number www.mathsrevision.com Q1 Q2 Q3 The quartiles fall in the gaps between Q1 : the 2th number Q2 : the 4th number Q3 : the 6th number. 3 4 10 Created by Mr Lafferty Maths Dept

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

28. S3 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

29. Quartiles fromCumulative FrequencyGraphs S3 Credit 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.

30. Quartiles fromCumulative FrequencyGraphs S3 Credit www.mathsrevision.com

31. Interquartile Range = (36 - 21) = 15 Semi-interquartile range SIQR = (Q3 – Q1 )÷2 = (36 - 21)÷2 =7.5 Cumulative FrequencyGraphs S3 Credit Quartiles 40 ÷ 4 =10 Q3 Q3 =36 Q2 Q2 =27 www.mathsrevision.com Q1 Q1 =21

32. Quartiles fromCumulative FrequencyGraphs S3 Credit www.mathsrevision.com

33. Interquartile range = (37 - 28) = 9 Semi-interquartile range = (Q3 – Q1 ) ÷2 = (37 - 28) ÷2 = 4.5 Cumulative FrequencyGraphs Cumulative FrequencyGraphs S3 Credit Q3 = 37 Quartiles 80 ÷ 4 =20 Q2 = 32 www.mathsrevision.com Q1 =28

34. Statistics Working Out Statistics Now try Ex 5.2 Ch12 (page 238) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

35. S3 Credit Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

36. Standard Deviation S3 Credit Learning Intention Success Criteria • Know the term Standard Deviation. • 1. To explain the term and calculate the Standard Deviation for a collection of data. • 2. Calculate the Standard Deviation for a collection of data. www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

37. 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. 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.

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

39. 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 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.

40. 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 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.

41. Standard Deviation For a FULL set of Data 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.

42. Relative Frequencies Now try Ex 6.1 Ch12 (page 240) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

44. Standard Deviation For a Sample of Data We will use this version because it is easier to use in practice ! 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.

45. 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 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

46. Q1b(i) Calculate the mean : 720 ÷ 8 = 90 Q1b(ii) Calculate the sample deviation Standard 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 6400 6561 6889 www.mathsrevision.com 8100 8836 9216 9216 10000 Created by Mr. Lafferty Maths Dept. ∑x = 720 ∑x2 = 65218

47. 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 Athletes Staff www.mathsrevision.com Created by Mr. Lafferty Maths Dept.

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

49. S3 Credit Starter Questions www.mathsrevision.com 33o Created by Mr. Lafferty Maths Dept.

50. S3 Credit 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