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Mr Barton’s Maths Notes Stats and Probability 7. Bar Charts and Histograms www.mrbartonmaths.com 7. Bar Charts and Histograms • The answer to this question is similar to the one for: “why do we bother working out averages Why do we bother with Statistical Diagrams? and measures of spread?”. • We live in a world jam-packed full of statistics, and if we were forced to look at all the facts and figures in their raw, untreated form, not only would we probably not be able to make any sense out of them, but there is also a very good chance our heads would explode. • Statistical Diagrams – if they are done properly - present those figures in a clear, concise, visually pleasing way, allowing us to make some sense out of the figures, summarise them, and compare them to other sets of data. Big Example To the right is a table showing the length of applause after Mr Barton announces that there will be no homework tonight. Construct a Bar Chart and a Histogram, and comment on the differences Length of Applause (mins) Frequency 0 < a ≤ 1 2 1 < a ≤ 2 4 2 < a ≤ 3 15 3 < a ≤ 5 10 5 < h ≤ 8 6 8 < h ≤ 13 5 1. Drawing a Bar Chart (Frequency Diagram) Note: Sometimes bar charts are called Frequency Diagrams! 1. Decide on an appropriate scale to fit the paper you are working with – as a general rule, the bar chart (or any statistical diagram, for that matter) should take up between half and three-quarters of the space you have to work with. Crucial: Your numbers must go up in equal steps!... see 8. Scatter Diagrams for examples of some very dodgy scales! 2. Label your axes. Is it usual to put frequency (total) on the y axis, and whatever the data is along the x axis. 3. Carefully draw in your bars… and add a title! Note: In examples like this where the groups are numbers, then it is usual to have the bars touching each other If the groups were categories (such as “colour of cars”), then you could have gaps between your bars if you like! A Bar Chart to show the Length of Applause after Mr Barton says “no homework” 20 frequency 18 16 14 12 10 8 6 4 2 applause (mins) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 2. Drawing a Histogram The major difference between a bar chart and a histogram is what goes on the y axis On a Bar Chart it is Frequency On a Histogram it is… Frequency Density! The reasons for this will be discussed soon! 1. Add two extra columns to your table… Group Width and Frequency Density 2. To work out Group Width, we just do the upper limit of each group minus the lower limit 3. To work out Frequency Density, we use this lovely formula: Frequency Frequency Density = Group Width 4. We then plot frequency density on the y axis, and our data on the x axis as before Note: In Histograms you have no choice… the bars must always be touching! Group Width: 1 – 0 = 1 Frequency Density: 2÷1=2 Length of Applause (mins) Frequency Group Width Frequency Density 0 < a ≤ 1 2 1 2 1 < a ≤ 2 4 1 4 2 < a ≤ 3 15 1 15 3 < a ≤ 5 10 2 5 5 < h ≤ 8 6 3 2 8 < h ≤ 13 5 5 1 Group Width: 13 – 8 = 5 Frequency Density: 5÷5=1 A Histogram to show the Length of Applause after Mr Barton says “no homework” 20 frequency density 18 16 14 12 10 8 6 4 2 applause (mins) 1 2 3 Note: Sometimes you get asked to draw a Frequency Polygon. Do not fear, this is just a Histogram, but with the top mid-points of each bar joined together with straight lines! 4 5 20 6 7 8 9 10 11 12 13 14 15 frequency density 18 16 14 12 10 8 6 4 2 applause (mins) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 3. What is the Point of Histograms? • • Have a quick look back at the two diagrams On the Bar Chart, which group looks like it has the most people in it?… maybe the last one?... because it takes up such a large area compared to the other groups… but if you look on the table of data, this group only had a total of 5! So… Bar Charts can be deceptive • • • But look at the Histogram…the group that looks the biggest now is the “between 2 and 3 minutes” group… and this is the group that has the highest frequency! The reason for this is that in a Histogram, the areas of the bars are proportional to the frequency, and not just the height like a bar chart. Note: If all our groups had the same width, it wouldn’t matter, but often they do not, so that is why Histograms tend to be used more than Bar Charts! 4. Interpreting Histograms Example: Here is a Histogram showing the time taken by some year 7s to complete all their times tables. Find the frequencies of each group. 20 frequency density 18 16 14 12 10 8 6 4 2 time (mins) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 This time we are given Frequency Density, and asked to work out Frequency… well, if we do a little re-arranging to our formula we get… Frequency = Frequency Density x Group Width Time (mins) Frequency Density Group Width Working Frequency 0 < t ≤ 3 2 3 2 x 3 = 6 6 3 < t ≤ 5 4 2 4 x 2 = 8 8 5 < t ≤ 6 10 1 10 x 1 = 10 10 6 < t ≤ 8 5 2 5 x 2 = 10 10 8 < t ≤ 12 1.5 4 1.5 x 4 = 6 6 Good luck with your revision!