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HISTOGRAMS

HISTOGRAMS. Representing Data. Why use a Histogram. When there is a lot of data When data is Continuous a mass, height, volume, time etc Presented in a Grouped Frequency Distribution Often in groups or classes that are UNEQUAL . NO GAPS between Bars. Histograms look like this.

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HISTOGRAMS

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  1. HISTOGRAMS Representing Data

  2. Why use a Histogram • When there is a lot of data • When data is • Continuous • a mass, height, volume, time etc • Presented in a Grouped Frequency Distribution • Oftenin groups or classes that are UNEQUAL

  3. NO GAPS between Bars Histograms look like this...... Continuous data

  4. Bars may bedifferent in width Determined by Grouped Frequency Distribution

  5. So we use FREQUENCY DENSITY AREA is proportional to FREQUENCY = Frequency Class width NOT height, because of UNEQUAL classes!

  6. Grouped Frequency Distribution Classes These classes are well defined there are no gaps !

  7. Drawing • Sensible Scales • Bases of rectangles correctly aligned • Plot the Class Boundaries carefully • Heights of rectangles needs to be correct • Frequency Density

  8. Frequency Densities Class width 40 10 10 30 20 2.0 1.5 2.5 3.0 1.5

  9. 3.0 2.0 1.0 0 20 40 60 80 100 120 Frequency = Width x Height Freq Dens Frequency = 40 x 2.0 = 80 Speed (km/h)

  10. Grouped Frequency Distribution GAPS! Need to adjust to Continuous Classes No gaps Ready to graph

  11. 4½ 9½ 19½ 29½ 39½ 59½ 5 10 10 10 20 Adjusting Classes Class Widths

  12. Frequency Density

  13. Drawing • Sensible Scales • Bases correctly aligned • Plot the Class Boundaries • Heights correct • Frequency Density

  14. 3.0 2.0 1.0 4.5 9.5 19.5 29.5 39.5 49.5 59.5 Freq Dens Time (Mins) 5 10 15 20 25 30 35 40 45 50 55 60

  15. Estimating a Frequency • Imagine we want to Estimate the number of people with a time between 12 and 25 mins • Because we have rounded to nearest minute with our classes we......... • Consider the interval from 11.5 to 25.5

  16. Freq Dens 11.5 25.5 3.0 2.0 1.0 4.5 9.5 19.5 29.5 39.5 49.5 59.5 Time (Mins) Width FD Frequency = 0.9 x 8 = 7.2 Frequency = 1.8 x 6 = 10.8 Total Frequency = 18

  17. We can estimate the Mode Mode is therefore in this Class

  18. Freq Dens 3.0 2.0 1.0 4.5 9.5 19.5 29.5 39.5 49.5 59.5 Time (Mins) Modal class

  19. …and the other one? • Simpler to plot • No adjustments required – class widths friendly • No ½ values • Estimation from the EXACT values given • No adjustment required • Estimate 15 to 56 would use 15 and 56! • Appear LESS OFTEN in the exam

  20. Why use frequency density for the vertical axes of a Histogram? • The effect of unequal class sizes on the histogram can lead to misleading ideas about the data distribution The vertical axis is Frequency Density

  21. Example: Misprediction of Grade Point Average (GPA)The following table displays the differences between predictedGPA and actual GPA. Positive differences result when predictedGPA > actual GPA. 17.1% of data X 10-3 2.3% of data The frequency histogram considerably exaggerates the incidence of overpredictedand underpredictedvalues The area of the two most extreme rectangles are much too large.!! 1000

  22. Example: Density Histogram of Misreporting GPA To avoid the misleading histogram like the one on last slide, display the data with frequency density Frequency=( rectangle height )x( class width ) = area of rectangle

  23. X 10-3 Frequency density x 10-3

  24. Principles of Excellent Graphs The graph should not distort the data. The graph should not contain unnecessary things (sometimes referred to as chart junk). The scale on the vertical axis should begin at zero. All axes should be properly labelled. The graph should contain a title. The simplest possible graph should be used for a given set of data. Chap 2-24

  25. Graphical Errors: Chart Junk Bad Presentation Minimum Wage 1960: $1.00 1970: $1.60 1980: $3.10 1990: $3.80  Good Presentation Minimum Wage $ 4 2 0 1960 1970 1980 1990 Chap 2-25

  26. Graphical Errors: No Relative Basis  Bad Presentation Good Presentation A’s received by students. A’s received by students. % Freq. 30% 300 20% 200 10% 100 0 0% FD UG GR SR FD UG GR SR FD = Foundation, UG = UG Dip, GR = Grad Dip, SR = Senior Chap 2-26

  27. Graphical Errors: Compressing the Vertical Axis  Bad Presentation Good Presentation Quarterly Sales Quarterly Sales $ $ 50 200 25 100 0 0 Q1 Q2 Q4 Q3 Q4 Q1 Q2 Q3 Chap 2-27

  28. Graphical Errors: No Zero Point on the Vertical Axis  Bad Presentation Good Presentations $ Monthly Sales Monthly Sales $ 45 45 42 42 39 39 36 36 0 J F M A M J J M A M J F Graphing the first six months of sales Chap 2-28

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