Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com GCSE: Histograms Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Objectives: To understand why a histogram is useful for displaying data, and how to both draw and interpret a histogram. Last modified: 13th January 2016

Click to Start Bromanimation Pablo is hosting a party. He counts how many people are between 15 and 20, and 20 and 50. Age (years) Frequency 15 ≤ a < 20 15 20 ≤ a < 50 Why is below graph somewhat unhelpful. How could we fix it? Click to Start Bromanimation 15 Frequency 10 20 30 40 50 Age

Click to Start Bromanimation 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. Age (years) Frequency 15 ≤ a < 20 15 20 ≤ a < 50 ? ? Click to Start Bromanimation 3 2 1 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 Frequency Density Estimated Frequency ? 10 20 30 40 50 Age

Bar Charts vs Histograms 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. ? ? ? ? Frequency Density Frequency 1.0m 1.2m 1.4m 1.6m 1.8m 6 7 8 9 Height Shoe Size

Copy and complete ? ? ? ? ? ? ? ? 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 ? ? Freq ? F.D. Width ? Frequency = 40 ? 5 4 3 2 1 Frequency = 15 ? Frequency = 25 ? Frequency Density Frequency = 30 ? 10 20 30 40 50 Height (m)

? ? ? ? ? ? ? The Box of Helpfulness Freq F.D. Width Start by adding a Frequency Density column Frequency Density 30  30 = 1 ? 84 ? 4.2 (using graph) ? ? 60 6 (using graph) ? 40  20 = 2 ? 18  30 = 0.6 ? 1 2 3 4 5 6 7 8 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? This triangle will help throughout. Freq F.D. Width

Determining the frequency density scale Copy the diagram and table, then work out the scale on the frequency density axis. 4 3 2 1 ? ? 16 12 8 4 ? 2 1 Frequency Density Frequency Density Frequency Density 0 10 20 0 10 20 20 28 36 Height (m) 0≤𝑥<15 Frequency 30 Height (m) 0≤𝑥<5 Frequency 60 Height (m) 20≤𝑥<32 Frequency 6

Questions Provided collection of past GCSE questions.

Summary ? ? ? ? ? Tips you might give your classmates... Purpose: Histograms allow us to display continuous data grouped into (potentially non-fixed) intervals. 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: ? ? Working out proportion of things (no FD scale given): Use any arbitrary scale for FD axis. Use it to find area of region that matches description. Divide by total area. Freq ? F.D. Width

Proportion Histogram Questions Sometimes you have to find the proportion of people/things/animals within some range of values. Just find the total area, and the area you’re interested in. 8 7 6 5 4 3 2 1 Total area =50 ? What proportion of people had a height: Between 10 and 14m: 𝟏𝟔 𝟓𝟎 = 𝟖 𝟐𝟓 Between 14 and 18m: 𝟏𝟖 𝟓𝟎 = 𝟗 𝟐𝟓 ? Frequency Density ? 10 14 18 22 26 Bro Tip: If the frequency density scale is missing, you can set it to what you like. Height (m)

Solutions – Questino 2 ? 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 = 32.4 44.4 =𝟎.𝟕𝟑 ? This is Q2 on your worksheet.

Questions Provided collection of past GCSE questions.

Solutions – Question 1 Answer: 17.24 57 ×285=86.2 ?

Solutions – Question 3 FD 16 4 2.4 4.8 16 12 8 4 Frequency Density 40 60 56 32 ? B1 for Frequency density label or appropriate units B2 for 4 correct histogram bars sq (B1 for 2 bars correct)

Solutions – Question 4 ? 0.05 0.04 0.03 0.02 0.01 8 6 ?

Solutions – Question 5