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GCSE - Histograms

ζ. Dr Frost. GCSE - Histograms. Objectives: To understand why a histogram is useful for displaying data, and how to both draw and interpret a histogram. 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.

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GCSE - Histograms

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  1. ζ Dr Frost GCSE - Histograms Objectives: To understand why a histogram is useful for displaying data, and how to both draw and interpret a histogram.

  2. 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 15 Frequency 10 20 30 40 50 Age

  3. 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 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 Estimated Frequency Frequency Class Width ? Frequency Density = 10 20 30 40 50 Age

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

  5. Copy and complete ? ? 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)

  6. 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 ? 8 The Box of Helpfulness 7 We don’t know the scale on the frequency density axis. Can we work it out using the first row of the table? 6 5 4 3 2 This triangle will help throughout. Freq 1 F.D. Width

  7. Determining the frequency density scale Copy the diagram and table, then work out the scale on the frequency density axis. ? 4 3 2 1 ? 2 1 ? 16 12 8 4 Frequency Density Frequency Density Frequency Density 0 10 20 20 28 36 0 10 20

  8. Exercises Provided collection of past GCSE questions.

  9. In pairs, work out the following… 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 = ?

  10. Summary Tips you might give your classmates... 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. Purpose: Histograms allow us to display continuous data grouped into (potentially non-fixed) intervals. ? ? 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

  11. More Difficult 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 ? What proportion of people had a height: Between 10 and 14m: Between 14 and 18m: ? Frequency Density ? Bro Tip: If the frequency density scale is missing, you can set it to what you like. 10 14 18 22 26 Height (m)

  12. More Difficult Histogram Questions Answer: ?

  13. More Difficult Histogram Questions Bro Tip: The total area represents the total frequency. Thus to find the frequency of some portion of the histogram, just times the total frequency by the fraction of the histogram that you’re interested in. Bro Tip: When the frequency density scale is not given, you can make it what you like! Answer: ?

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