Edexcel IGCSE-Drawing and Interpretation Histograms.pptx

These slides are used for GCSE students.

1. GCSE: Histograms Dr J Frost ([email protected]) www.drfrostmaths.com Last modified: 4 th October 2020 Objectives: To understand why a histogram is useful for displaying data, and how to both draw and interpret a histogram.

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3. Age (years) Frequency 15 ≤ a < 20 15 20 ≤ a < 50 15 10 20 30 40 50 Age Frequency 15 Pablo is hosting a party. He counts how many people are between 15 and 20, and 20 and 50. Why is below graph somewhat unhelpful. How could we fix it? Click to Start Bromanimation The 15 people in the second group are more spread out in age, but this graph seems to suggest that people’s ages are spread out uniformly between 15 and 50.

4. Age (years) Frequency 15 ≤ a < 20 15 20 ≤ a < 50 15 10 20 30 40 50 Age Estimated Frequency 3 2 1 Let’s presume that within each age group, the ages are evenly spread. Then there would 3 people of each age in the 15-20 group, and 0.5 people of each age in the 20-50 group. Click to Start Bromanimation ? ? Frequency Density The resulting diagram is known as a histogram . The ‘frequency per age’ is known as the ‘ frequency density ’. In general, given the frequency and class width, we can calculate it using: Frequency Density = Frequency Class Width ?

5. 6 7 8 9 Shoe Size Frequency Height 1.0m 1.2m 1.4m 1.6m 1.8m Frequency Density Bar Charts For discrete data. Frequency given by height of bars. Histograms For continuous data. Data divided into (potentially uneven) intervals. Frequency given by area of bars. ? ? ? ? Bar Charts vs Histograms

6. F.D. Freq Width Weight ( w kg) Frequency Frequency Density 0 < w ≤ 10 40 4 10 < w ≤ 15 6 1.2 15 < w ≤ 35 52 2.6 35 < w ≤ 45 10 1 ? ? ? ? 10 20 30 40 50 Height (m) 5 4 3 2 1 Frequency Density Frequency = 15 Frequency = 30 Frequency = 40 Frequency = 25 ? ? ? ? 1 2 Examples

7. F.D. Freq Width This triangle will help throughout. The Box of Helpfulness We don’t know the scale on the frequency density axis. Can we work it out using the first row of the table? 1 2 3 4 5 6 7 8 84 60 ? ? 40  20 = 2 18  30 = 0.6 ? ? 30  30 = 1 ? Frequency Density Always start by adding a Frequency Density column 4.2 (using graph) ? 6 (using graph) ? Bro Tip : For this kind of question, first find a ‘complete’ set of information: i.e. for the first row in this question, we have the frequency and the drawn bar, so can work out the F.D. scale. 3

8. Test Your Understanding (on your sheet) Bro Hint: The second row has a ‘complete’ set of information (bar and frequency) FD 3.8 4.8 4.2 3.1 0.8 4 3 2 1 5 31 ? ?

9. Frequency Density 0 10 20 Height (m) Frequency Height (m) Frequency 4 3 2 1 ? Work out the scales on the frequency density axis. Frequency Density 0 10 20 Height (m) Frequency Height (m) Frequency 16 12 8 4 ? Frequency Density 20 28 36 Height (m) Frequency Height (m) Frequency 2 1 ? Quickfire Questions – Determining F.D. scale

10. Provided collection of past GCSE questions. (Answers on next slides) Questions

11. Question 1 FD 3.2 2 1.2 0.6 0.4 3.2 2.4 1.6 0.8 4.0 12 6 ? ?

12. Question 2 2.0 1.5 1.0 0.5 FD 0.8 1 1.6 2 1.2 ? ?

13. Question 3 FD 3 5 3.6 1.2 4 3 2 1 5 ? (b) Work out an estimate for the number of cars with a speed of more than 85 km/h. Note that 85 to 100 is three-quarters of the 80 to 100 interval. Thus we can estimate we’ll have of the 24 drivers in this group.   ?

14. Question 4 FD 1 3.5 3 1.5 0.5 4 3 2 1 30 60 ? ? ?

15. Question 5 24 ? 30 ?

16. Purpose: Histograms allow us to display continuous data grouped into ( potentially non-fixed ) intervals. The idea is that they reflect the ‘concentration’ of things within each range of values. Area: The area of a bar is equal to the frequency*. * Actually it’s only proportional to it, but you don’t need to worry about that till A Level. Working out the F.D. scale: If the frequency is known and the bar height is known, we can work out the scale using the formula on the left. Frequency Density Formula: Frequency Density is ‘frequency per unit value’, i.e : F.D. Freq Width ? ? ? ? Summary So Far

17. Harder Histogram Questions We previously saw that the area gives us frequency. If we don’t have the frequency density axis, we could use the idea that area and frequency are in proportion to each other . 130cm-135cm  20 squares Each square worth: children 110-130cm  70 squares children   One strategy: Since area represents frequency, find out how many people each square is worth. ? If each square is 0.5 children, we need 12 squares. Click to show?

18. Harder Histogram Questions So finding area between 110cm and 130cm (as area = frequency)   However, there’s nothing stopping you using the same approach as before: working out the frequency density axis from a ‘complete’ set of information (i.e. where we have the frequency AND the bar). ? 1 2 3

19. Test Your Understanding ( use either method ) (If using square count method, perhaps count big squares rather than little squares.) 18 big squares 9 students So 1 square = 0.5 students Total squares = 30 15 students OR: F.D. = Then use areas of bars.   a b 24 (big) squares required (We can have 4 rows of 5 to get first 20 squares, but have to split the last 4 squares across 5, thus use of a square for each square at top) This is quite a clumsy method, so the frequency density approach would probably be easier here.   ? ?

20. Proportion Histogram Questions Sometimes you have to find the proportion of people/things/animals within some range of values. Frequency Density Height (m) 10 14 18 22 26 What proportion of people had a height: Between 10 and 14m: Between 14 and 18m:   Total area   Bro Tip: If the frequency density scale is missing, you can set it to what you like for this kind of question. ? ? ? 8 7 6 5 4 3 2 1 Key Point : Since area is frequency, the proportion of the area is the same as the proportion of the frequency. e.g. If half the area is above 18m, then half the people are above 18m.

21. Solution: Total apples: (40 x 0.12) + (20 x 0.36) + (20 x 0.7) + (20 x 0.56) + (40 x 0.18) = 44.4 Apples in range 140-160g: (20 x 0.36) + (20 x 0.7) + (20 x 0.56) = 32.4 Proportion =   ? Test Your Understanding Bro Tip : I recommend writing the area of each bar on the bar itself, as you’ll likely have to use some areas twice.

22. Provided collection of past GCSE questions. Questions

23. Solutions – Question 1 If we use 1,2,3,… on frequency density scale: Total area Area between 25 and 40 mins: Percentage:   a b of area is between 10-15 mins.   ? ?

24. Solutions – Question 2 Answer:   ? 10 20 30 40 50 60

25. Solutions – Question 3 The histogram shows information about the lifetime of some batteries. Two of the batteries had a lifetime of between 1.5 and 2.5 years. Find the total number of batteries. Total batteries batteries   2 4 6 8 10 12 14 ?

26. Solutions – Question 4 40 60 56 32 FD 16 4 2.4 4.8 16 12 8 4 Frequency Density B1 for Frequency density label or appropriate units B2 for 4 correct histogram bars sq (B1 for 2 bars correct) ?

27. Solutions – Question 5 8 6 ? 0.05 0.04 0.03 0.02 0.01 ?