2. Chapter 2 Frequency Distributions, Stem-and-leaf displays, and Histograms

3. Where have we been?

4. To calculate SS, the variance, and the standard deviation: find the deviations from , square and sum them (SS), divide by N (  2 ) and take a square root(  ). Example: Scores on a Psychology quiz Student John Jennifer Arthur Patrick Marie X78357X78357  X = 30 N = 5  = 6.00 X -  +1.00 +2.00 -3.00 +1.00  (X-  ) = 0.00 (X -  ) 2 1.00 4.00 9.00 1.00  (X-  ) 2 = SS = 16.00  2 = SS/N = 3.20  = = 1.79

5. Ways of showing how scores are distributed around the mean zFrequency Distributions, zStem-and-leaf displays z Histograms

6. Some definitions zFrequency Distribution - a tabular display of the way scores are distributed across all the possible values of a variable zAbsolute Frequency Distribution - displays the count of each score. zCumulative Frequency Distribution - displays the total number of scores at and below each score. zRelative Frequency Distribution - displays the proportion of each score. zRelative Cumulative Frequency Distribution - displays the proportion of scores at and below each score.

7. Example Data Traffic accidents by bus drivers Studied 708 bus drivers. Recorded all accidents for a period of 4 years. Data looks like: 3, 0, 6, 0, 0, 2, 1, 4, 1, … 6, 0, 2

8. Frequency Distributions # of accidents 0 1 2 3 4 5 6 7 8 9 10 11 Absolute Freq. 117 157 158 115 78 44 21 7 6 1 3 1 708 Relative Frequency.165.222.223.162.110.062.030.010.008.001.004.001.998 Calculate relative frequency. Divide each absolute frequency by the N. For example, 117/708 =.165 Notice rounding error

9. What can you answer? # of accidents 0 1 2 3 4 5 6 7 8 9 10 11 Relative Freq..165.222.223.162.110.062.030.010.008.001.004.001.998 Proportion with at most 1 accident? Proportion with 8 or more accidents? =.165 +.222 =.387.387 * 100 = 38.7% =.008 +.001 +.004 +.001 =.014 = 1.4% Proportion with between 4 and 7 accidents? =.110 +.062 +.030 +.010 =.212 = 21.2%

10. Cumulative Frequencies # of acdnts 0 1 2 3 4 5 6 7 8 9 10 11 Absolute Frequency 117 157 158 115 78 44 21 7 6 1 3 1 708 Cumulative Frequency 117 274 432 547 625 669 690 697 703 704 707 708 Cumulative Relative Frequency.165.387.610.773.883.945.975.983.993.994.999 1.000 Cumulative frequencies show number of scores at or below each point. Calculate by adding all scores below each point. Cumulative relative frequencies show the proportion of scores at or below each point. Calculate by dividing cumulative frequencies by N at each point.

11. Grouped Frequency Example 100 High school students’ average time in seconds to read ambiguous sentences. Values range between 2.50 seconds and 2.99 seconds.

12. Grouped Frequencies Needed when ynumber of values is large OR yvalues are continuous. To calculate group intervals yFirst find the range. yDetermine a “good” interval based on xon number of resulting intervals, xmeaning of data, and xcommon, regular numbers. yList intervals from largest to smallest.

13. Grouped Frequencies Reading Time 2.90-2.99 2.80-2.89 2.70-2.79 2.60-2.69 2.50-2.59 Reading Time 2.95-2.99 2.90-2.94 2.85-2.89 2.80-2.84 2.75-2.79 2.70-2.74 2.65-2.69 2.60-2.64 2.55-2.59 2.50-2.54 Frequency 16 31 20 12 21 Frequency 9 7 20 11 10 4 8 10 11 Range = 2.99 - 2.50 =.49 ~.50 i =.1 #i = 5 i =.05 #i = 10

14. Either is acceptable. zUse whichever display seems most informative. zIn this case, the smaller intervals and 10 category table seems more informative. zSometimes it goes the other way and less detailed presentation is necessary tp prevent the reader from missing the forest for the trees.

15. Stem and Leaf Displays zUsed when seeing all of the values is important. zShows ydata grouped yall values yvisual summary

16. Stem and Leaf Display zReading time data Reading Time 2.9 2.8 2.7 2.6 2.5 Leaves 5,5,6,6,6,6,8,8,9 0,0,1,2,3,3,3 5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,9,9,9,9 0,0,1,2,3,3,3,3,4,4,4 5,5,5,5,6,6,6,8,9,9 0,0,0,1,2,3,3,3,4,4 5,6,6,6 0,1,1,1,2,3,3,4 6,6,8,8,8,8,8,9,9,9 0,1,1,1,2,2,2,4,4,4,4 i =.05 #i = 10

17. Stem and Leaf Display zReading time data Reading Time 2.9 2.8 2.7 2.6 2.5 Leaves 0,0,1,2,3,3,3,5,5,6,6,6,6,8,8,9 0,0,1,2,3,3,3,3,4,4,4,5,5,5,5,5,6,6,6,7,7,7,7,7,7,7,8,9,9,9,9 0,0,0,1,2,3,3,3,4,4,5,5,5,5,6,6,6,8,9,9 0,1,1,1,2,3,3,4,5,6,6,6 0,1,1,1,2,2,2,4,4,4,4,6,6,8,8,8,8,8,9,9,9 i =.1 #i = 5

18. Transition to Histograms 999977777776665555999977777776665555 988666655988666655 33321003332100 4443333210044433332100 99866655559986665555 44333210004433321000 66656665 4332111043321110 4444222111044442221110 99988888669998888866

19. Histogram of reading times 20 18 16 14 12 10 8 6 4 2 0 Reading Time (seconds) FrequencyFrequency

20. Histogram concepts - 1 zUsed to display continuous data. zDiscrete data are shown on a box graph. zBut most psychology data are continuous, even if they are measured with integers.

21. Histogram concepts - 2 zUse bar graphs, not histograms, for discrete data. zYou rarely see data that is really discrete. zDiscrete data are categories or rankings. zIf you have continuous data, you can use histograms, but remember real class limits. zHistograms can be used for relative frequencies as well.

22. What are the real limits of each class? 20 18 16 14 12 10 8 6 4 2 0 Real limits of the fifth class are ???? - ???? Real limits of the highest class are ???? - ????. FrequencyFrequency

23. What are the real limits of each class? 20 18 16 14 12 10 8 6 4 2 0 Real limits of the fifth class are 2.695-2.745 Real limits of the highest class are 2.945 - 2.995 FrequencyFrequency

24. Predicting from Theoretical Distributions zTheoretical distributions show how scores can be expected to be distributed around the mean. (Mean = 2.755 for reading data). zDistributions are named after the shapes of their histograms: yRectangular yJ-shaped yBell (Normal) ymany others

25. Rectangular Distribution of scores

26. Flipping a coin 100 flips - how many heads and tails do you expect? Heads Tails 100 75 50 25 0

27. Rolling a die 120 rolls - how many of each number do you expect? 1 2 3 4 5 6 100 75 50 25 0

28. Rolling 2 dice How many combinations are possible? Dice Total 1 2 3 4 5 6 7 8 9 10 11 12 Absolute Freq. 0 1 2 3 4 5 6 5 4 3 2 1 36 Relative Frequency.000.028.056.083.111.139.167.139.111.083.056.028 1.001

29. Rolling 2 dice 360 rolls - how many of each number do you expect? 1 2 3 4 5 6 7 8 9 10 11 12 100 90 80 70 60 50 40 30 20 10 0

30. Normal Curve

31. J Curve Occurs when socially normative behaviors are measured. Most people follow the norm, but there are always a few outliers.

32. Principles of Theoretical Curves zExpected frequency = Theoretical relative frequency * N zExpected frequencies are your best estimates because they are closer, on the average, than any other estimate when we square the error. zLaw of Large Numbers - The more observations that we have, the closer the relative frequencies should come to the theoretical distribution.

33. Q & A: Continuous data zHOW IS THE FACT THAT WE ARE DISPLAYING CONTINUOUS DATA SHOWN ON A HISTOGRAM AS OPPOSED TO A BAR GRAPH? zThe bars of the graph on a histogram meet at the real limits of each interval. zIF DATA CAN ONLY BE INTEGERS (SUCH AS NUMBER OF TRUE/FALSE QUESTIONS ANSWERED CORRECTLY ON A PSYCH QUIZ), HOW COME IT IS CALLED CONTINUOUS DATA. zWhether data is continuous or discrete depends on what your measuring, not the accuracy of your measuring instrument. For example, distance is continuous whether you measure it with a yardstick or a micrometer. Knowledge, like self-confidence and other psychological variables, is probably best thought of as a continuous variable.

34. Determining “i” (the size of the interval) zWHAT IS THE RULE FOR DETERMINING THE SIZE OF INTERVALS TO USE IN WHICH TO GROUP DATA? zWhatever intervals seems appropriate to most informatively present the data. It is a matter of judgement. Usually we use 6 – 12 same size intervals each of which use intuitively obvious endpoints (e.g., 5s and 0s).