1. S1: Chapters 2-3 Data: Location and Spread Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 5 th September 2014

2. Types of variables In statistics, we can use a variable to represent some quantity, e.g. height, age. This could be qualitative (e.g. favourite colour) or quantitative (i.e. numerical). 2 types of variable: Discrete variables Has specific values. e.g. Shoe size, colour, website visits in an hour period, number of siblings, … Continuous variables Can have any value in a range. e.g. Height, distance, weight, time, wavelength, … ??

3. 5 th ? Quartiles for large numbers of items LQMedianUQ 31 8 th 16 th 24 th 19 10 th 15 th 6 3 rd and 4 th 2 nd 5 th 14 7 th and 8 th 4 th 11 th ??? ?? ?? ? ? ? ? Under what circumstances do we not round? When we have a grouped frequency table involving a continuous variable. ?

4. 3 2.51.53.5 1.524.5 23.55 21 ??? ??? ??? ??? Quickfire Quartiles 1, 2, 3 LQMedianUQ 1, 2, 3, 4 1, 2, 3, 4, 5 1, 2, 3, 4, 5, 6

5. Notation for quartiles/percentiles Lower Quartile: Median: Upper Quartile: 57 th Percentile: ? ? ? ?

6. Grouped Frequency Data Recap Why is our mean just an estimate? The midpoints of each interval. They‘re effectively a sensible single value used to represent each interval. Because we don’t know the exact heights within each group. Grouping data loses information. ??? ? ? ? Frequency This type of data is continuous. ?

7. Grouped Frequency Data Recap Frequency Modal class interval: Median class interval: (‘modal’ means ‘most’) There are 40 items, so determine where 20 th item is. ? ?

8. Using STATS mode on your calculator Frequency Warning: You still need to show working in the exam. Work out the mean for this example first using proper workings.

9. Weight of cat to nearest kgFrequency What’s different about the intervals here? There are GAPS between intervals! What interval does this actually represent? Lower class boundaryUpper class boundary Class width = 3 ? ?

10. Identify the class width … Class width = 10 ? … … … Lower class boundary = 200 ? Class width = 3 ? Lower class boundary = 3.5 ? Class width = 2 ? Lower class boundary = 29 ? Class width = 10 ? Lower class boundary = 30.5 ?

11. Interpolation S2 – Chapters 2/3

12. RECAP: Quartiles of Frequency Table Age of squirrelFrequencyCumulative Freq 155 2813 31124 4529 ? ? ?

13. Estimating the median Answer = 13.5 + 8 = 21.5 ? GCSE Question

14. Estimating the median At GCSE, you were only required to give the median class interval when dealing with grouped data. Now, we want to estimate a value within that class interval. Weight of cat to nearest kgFrequency ? 918 11 15.5kg?18.5kg Frequency up until this interval Frequency at end of this interval Item number we’re interested in. Weight at start of interval. Weight at end of interval. ? ? ? ? ? (Why not the 11.5 item?)

15. Estimating other values Weight of cat to nearest kgFrequency ? ? ?

16. You should have a sheet in front of you 1a 1b 1c 2a 2b 2c 1d ? ? ? ? ? ? ?

17. Exercises Page 34 Exercise 3A Q4, 5, 6 Page 36 Exercise 3B Q1, 3, 5

18. Variance and Standard Deviation S2 – Chapters 2/3

19. What is variance? Distribution of IQs in L6Ms4 Distribution of IQs in L6Ms5 Here are the distribution of IQs in two classes. What’s the same, and what’s different?

20. Variance Variance is how spread out data is. Variance, by definition, is the average squared distance from the mean. Distance from mean… Squared distance from mean… Average squared distance from mean…

21. Simpler formula for variance “The mean of the squares minus the square of the mean (‘msmsm’)” Variance ?? Standard Deviation The standard deviation can ‘roughly’ be thought of as the average distance from the mean.

22. Starter Calculate the variance and standard deviation of the following heights: 2cm3cm3cm5cm7cm ? ?

23. Practice Find the variance and standard deviation of the following sets of data. ?? ? ?

24. Extending to frequency/grouped frequency tables We can just mull over our mnemonic again: Variance: “The mean of the squares minus the square of the means (‘msmsm’)” ?? Bro Tip: It’s better to try and memorise the mnemonic than the formula itself – you’ll understand what’s going on better, and the mnemonic will be applicable when we come onto random variables in Chapter 8.

25. Example Frequency ? ?? ?

26. Sometimes we’re helpfully given summed data: Frequency ?

27. Exercises Page 40 Exercise 3C Q1, 2, 4, 6 Page 44 Exercise 3D Q1, 4, 5

28. Recap ? ? ? ?

29. Coding S2 – Chapters 2/3

30. Starter What do you reckon is the mean height of people in this room? Now, stand on your chair, as per the instructions below. INSTRUCTIONAL VIDEO Is there an easy way to recalculate the mean based on your new heights? And the variance of your heights?

31. Starter Suppose now after a bout of ‘stretching you to your limits’, you’re now all 3 times your original height. What do you think happens to the standard deviation of your heights? It becomes 3 times larger (i.e. your heights are 3 times as spread out!) ? What do you think happens to the variance of your heights? It becomes 9 times larger ? (Can you prove the latter using the formula for variance?)

32. The point of coding £1010 £1020 £1030 £1040 £1050 ? ? We ‘code’ our variable using the following: £1 £2 £3 £4 £5 ?

33. Finding the new mean/variance Old variance CodingNew variance ?? ?? ?? ?? ?? ??

34. Exercises Page 26 Exercise 2E Q3, 4 Page 47 Exercise 3E Q2, 3, 5, 7

35. Chapters 2-3 Summary For the following grouped frequency table, calculate: Frequency a) The estimate mean: b) The estimate median: ? ? ? ? ? ?

36. Chapters 2-3 Summary ? ? ? ? ?