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Frequency Distributions & Graphing

This guide covers the essential concepts of frequency distributions, including how to calculate and represent frequency (f) and percentage (P or %), and emphasizes the importance of accurate reporting. It introduces various graphical techniques for visualizing data, such as stem-and-leaf plots, boxplots, bar charts, histograms, and scatterplots. Each method provides unique insights into data distribution, trends, and relationships between variables. Practicing these graphical representations enhances data analysis skills in statistics.

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Frequency Distributions & Graphing

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  1. 1 Frequency Distributions & Graphing

  2. Nomenclature • Frequency: number of cases or subjects or occurrences • represented with f • i.e. f = 12 for a score of 25 • 12 occurrences of 25 in the sample 1

  3. Nomenclature • Percentage: number of cases or subjects or occurrences expressed per 100 • represented with P or % • So, if f = 12 for a score of 25 when n = 25, then... • % = 12/25*100 = 48% 1

  4. Caveat (Warning) • Should report the f when presenting percentages • i.e. 80% of the elementary students came from a family with an income < $25,000 • different interpretation if n = 5 compared to n = 100 • report in literature as • f = 4 (80%) OR • 80% (f = 4) OR 80% (n = 4) 1

  5. Frequency Distribution of Test Scores 2 3 4 • 40 items on exam • Most students >34 • skewed (more scores at one end of the scale) • Cumulative Percentage: how many subjects in and below a given score 1

  6. Eyeball check of data: intro to graphing with SPSS 1 • Stem and Leaf Plot: quick viewing of data distribution • Boxplot: visual representation of many of the descriptive statistics discussed last week • Bar Chart: frequency of all cases • Histogram: malleable bar chart • Scatterplot: displays all cases based on two values of interest (X & Y) • Note: compare to our previous discussion of distributions (normal, positively skewed, etc…) 2

  7. Stem and Leaf(SPSS: Explore command) 1 • Fast look at shape of distribution • shows f numerically & graphically • stem is value, leaf is f Frequency Stem & Leaf 2.00 Extremes (=<25.0) 2.00 28 . 00 2.00 29 . 00 1.00 30 . 0 1.00 31 . 0 3.00 32 . 000 1.00 33 . 0 6.00 34 . 000000 3.00 35 . 000 4.00 36 . 0000 8.00 37 . 00000000 Stem width: 1 Each leaf: 1 case 2 3 4

  8. Stem and Leaf Plots • Another way of doing a stemplot • Babe Ruth’s home runs in each of 14 seasons with the NY Yankees • 54, 59, 35, 41, 46, 25, 47, 60, 54, 46, 49, 46, 41, 34, 22 1 2 2 25 3 45 4 1166679 5 449 6 0 3

  9. Stem and Leaf Plots • Back-to-back stem plots allow you to visualize two data sets at the same time • Babe Ruth vs. Roger Maris MarisRuth 0 1 2 25 3 45 4 1166679 5 449 6 0 8 643 863 93 1 1

  10. Boxplots 1 Maximum Q3 Median Q1 Minimum Note: we can also do side-by-side boxplots for a visual comparison of data sets

  11. Format of Bar Chart Y axis (ordinate) 1 f X axis (abcissa) Individual scores/categories

  12. Test score data as Bar Chart Note only scores with non-zero frequencies are included. 1

  13. Bar chart in PASW • Using the height file on the web 2 1 3

  14. Bar chart in SPSS • Gives… 1 2

  15. Bar chart in PASW • Note you can use the same command for pie charts and histograms (next) 1

  16. Format of Histogram Now the X-axis is groups of scores, rather than individual scores – gives a better idea of the distribution underlying the data. Y axis (ordinate) f 1 X axis (abcissa) Can be manipulated Groups of scores/categories

  17. Test score data as Histogram 1

  18. Test score data as revised Histogram 1 With an altered number of groups, you might get a better idea of the distribution

  19. Scatterplot 1 2 3 • Quick way to visualize the data & see trends, patterns, etc… • This plot visually shows the relationship between undergrad GPA and GRE scores for applicants to our program 4

  20. Scatterplot 1 • Here’s the relationship between undergrad GPA (admitgpa) and GPA in our program

  21. Scatterplot 1 • Finally, here’s the relationship between GRE scores and GPA in our program

  22. Scatterplot in PASW 1 • Use graphs_scatter/Dot

  23. Scatterplot in PASW • Choose “simple scatter” 1

  24. Scatterplot in PASW • Choose the variables (here I’ve used a 3rd variable too – you’ll see why in a moment) 1

  25. Scatterplot in PASW 1 As you can see, there are rather different values for males and females

  26. Bottom line • First step should always be to plot the data and eyeball it...following is an example of what can happen when you do. 1

  27. One use of Frequency Distribution & Skewness 1 Expected distribution of agent-paid claims (State Farm) high low $$ amount

  28. One use of Frequency Distribution & Skewness 3 f Observed distribution of an agent-paid claims (hmmm…) 2 1 high low $$ amount

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