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

Where have we been?

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 -   (X-  ) = 0.00 (X -  )  (X-  ) 2 = SS =  2 = SS/N = 3.20  = = 1.79

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

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.

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

Frequency Distributions # of accidents Absolute Freq Relative Frequency Calculate relative frequency. Divide each absolute frequency by the N. For example, 117/708 =.165 Notice rounding error

What can you answer? # of accidents Relative Freq Proportion with at most 1 accident? Proportion with 8 or more accidents? = = * 100 = 38.7% = =.014 = 1.4% Proportion with between 4 and 7 accidents? = =.212 = 21.2%

Cumulative Frequencies # of acdnts Absolute Frequency Cumulative Frequency Cumulative Relative Frequency 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.

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.

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.

Grouped Frequencies Reading Time Reading Time Frequency Frequency Range = =.49 ~.50 i =.1 #i = 5 i =.05 #i = 10

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.

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

Stem and Leaf Display zReading time data Reading Time 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

Stem and Leaf Display zReading time data Reading Time 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

Transition to Histograms

Histogram of reading times Reading Time (seconds) FrequencyFrequency

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.

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.

What are the real limits of each class? Real limits of the fifth class are ???? - ???? Real limits of the highest class are ???? - ????. FrequencyFrequency

What are the real limits of each class? Real limits of the fifth class are Real limits of the highest class are FrequencyFrequency

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

Rectangular Distribution of scores

Flipping a coin 100 flips - how many heads and tails do you expect? Heads Tails

Rolling a die 120 rolls - how many of each number do you expect?

Rolling 2 dice How many combinations are possible? Dice Total Absolute Freq Relative Frequency

Rolling 2 dice 360 rolls - how many of each number do you expect?

Normal Curve

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

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.

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.

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).