Further Maths

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Further Maths

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Further Maths

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1. Further Maths Chapter 3 Summarising Numerical Data

2. The Mean : A measure of centre • The mean (often referred to as the average). • Mean = sum of values total number of values

3. The mean - Demo • Calculate the mean of the following numbers 4 6 7 8 9 11 = 45 = 6 7.5

4. The mean – Your Turn • Calculate the mean of the following numbers 14 20 25 30 32 133

5. The mean – from Stem Plot • 1 2 3 3 • 2 3 4 5 7 8 • 3 0 1 3 8 • 4 • 5 • 6 0

6. Median or Mean – Which one? mean = median = mode =

7. Median or Mean – Which one? mean = median = mode =

8. Median or Mean – Which one? mean = mode = median =

9. Median or Mean – Which one? • When the data is symmetrical and there are no outliers, either the mean or the median can be used to measure the centre • When the data is clearly skewed and/or there are outliers, the median should be used to measure the centre. • Exercise 3A Pages 59-60 Questions 1-8

10. The Standard Deviation : A measure of Spread about the Mean • To measure the spread of data about the mean, the standard deviation is used. • The formula for the standard deviation is • Calculate the standard deviation for the following numbers 4 6 7 8 9 11 • Calculator display 2

11. Estimating the Standard Deviation • The standard deviation is approximately equal to the range divided by 4. • Exercise 3B Pages 63-64 Questions 1 - 7

12. The normal distribution • Ball drop

13. The 68%, 95%, 99.7% rule for normal distributions

14. The 68%, 95%, 99.7% rule for normal distributions

15. The 68%, 95%, 99.7% rule for normal distributions

16. The 68%, 95%, 99.7% rule for normal distributions

17. The 68%, 95%, 99.7% rule for normal distributions

18. Example • The heights of 200 boys was found to be normally distributed with a mean of 180 cm and a standard deviation of 10 cm. • 68% of the heights will lie between • 95% of the heights will lie between • 99.7% of the heights will lie between

19. Example • What percentage of the heights will lie between 180 cm and 190 cm? • What percentage of the heights will lie below 200 cm? • What percentage of the boys will have a height between 160 cm and 190 cm? • How many boys should have a height above 190 cm? • Exercise 3C Pages 69-70 Questions 1-5

20. Z scores ( standardised scores) • Two students compare their scores for a test for two different subjects. One student scored 73 out of 100 for an English test. The other student scored 65 out of 100 for a Mathematics test. Which student performed better?

21. Z scores ( standardised scores)The plot thickens • Unbeknownst to the students the results for the English test were normally distributed with a mean of 80 and a standard deviation of 5. The results for the Mathematics test were normally distributed with a mean of 60 and a standard deviation of 8. Which student now performed better?

22. Z scores • Z scores are used to compare values from different distributions. A raw score, x, from a data set will have a z-score of (x – x ) s where x is the mean of the data set and s is the standard deviation. The closer the z score is to zero, the closer the raw score is to the mean.

23. Z scores

24. Z scores English z = (73 – 80) = - 7 = - 1.2 5 5 Maths z = (65 – 60) = 5 = 0.625 8 8

25. Questions • Exercise 3D Page 72 Questions 1-2

26. Generating a random sample • Draw numbers out of a hat • A spinner • Select every 2nd or 3rd person • Use technology

27. Using your calculator to generate a simple random sample • TI NSpire • Rand(25) will generate numbers from 1 to 25 • Calculator demonstration

28. Displaying and calculating the summary statistics for grouped data using a calculator