1. Frequency Distributions
Lecture 2:
Frequency Distributions

3. Exploratory Data Analysis
Emphasizes graphic representation
Addresses the following questions:
1. What is going on here?
2. Are there patterns in these data?
3. What are the patterns in these data?
Allows for hypothesis generation and model building through iteration
EDA is an attitude toward data - Tukey

5. Distributions
Distributions: arrangement of data; how the cases fall (are distributed)! Information about distributions is paramount in statistics.
Distributions can be displayed in tabular and graphic form
Frequency distributions: tabulation of the number of events/occurrences for each category on the scale of measurement
Frequency: a count of the number of occurrences

6. Frequency Table Pet type frequency proportion % dog cat fish turtle 3025 20 2 77
0.39
0.32
0.26
0.03
1.0 39 32 26 3
100

7. Grouping Data Arrange data from high to low
Frequencies can be used to find the total number of scores: f = N
X from a frequency distribution = fX
Proportion = p = f/N
Percentage = p (100) = f/N (100)
Score X frequency p % 15 10 4 20 11 46
0.43
0.33
0.24
1.0 43 33 24
100

8. X f p % 10 2 9 5 8 4 7 3 6
Finish filling the chart. Then find N andX

9. X f ΣX
Proportion % 10 2 20
.11 11 9 5 45
.28 28 8 4 32
.22 22 7 3 21
.17 17 6 12
N = 18
X = 140
1.0
100

10. Frequency Distributions
Ungrouped frequency distributions = a list of observations and their frequency occurrence when observations are sorted by single values.
Grouped frequency distributions = a list of observations and their frequency of occurrence when observations are sorted into categories or intervals

11. Grouped vs. ungrouped frequency distributions: how to decide
How many values?
If less than 10 different scores “ungrouped” is good
If more than 10 scores use grouped
Reminder: how to figure how many values
hi - lo + 1

12. Grouped Frequency Distributions
Crucial guidelines for constructing frequency distributions:
(1) aim for groups
(2) Mutually exclusive (each observation should be represented only once)
(3) All classes should have equal intervals (even if the frequency for a particular class is zero)
Wrong Right
(4) Don’t omit any intervals
(5) Make widths convenient (e.g. not 2.2) & bottom score should be a multiple of the width
Formula for finding the width in a grouped frequency distribution: interval = (hi - lo +1)/# groups

13. Example *Groups are called class intervals
i = ( )/ 10 = 3.2 ~ 3 midpoint = (hi real + lo real) /2
START at the Bottom with the low number
Interval Real Limits f Midpoint
*Groups are called class intervals
* Class intervals are apparent limits b/c it appears that they form the upper and lower boundaries for the class interval, but must take into account the real limits
NOTE: I violated the rule of making the bottom score a multiple of the width

14. Example * Include columns for interval, real limits, frequency,
* Include columns for interval, real limits, frequency,
and midpoint

15. Interval
Real limits f
Midpoint
92-95 4
93.5
88-91 5
89.5
84-87 6
85.5
80-83 1
81.5
76-79 9
77.5
72-75
73.5
68-71 8
69.5
64-67 7
65.5
60-63
61.5
56-59
57.5
52-55
53.5

16. Percentiles and Percentile Ranks: Get more out of your frequency distribution
Scores alone are meaningless. Compare a score to a standard score with percentiles.
Percentile: #s that divide the distribution into 100 = parts
Percentile rank: # that represents the % of cases in a comparison group that achieved scores  the one cited
e.g. PR of 95 on the SAT means that 95% of those taking the SAT performed equally or worse than you and only 5% did better

17. Example
Class grades f c f cprop cum % 32
First step: find the number of people located at or below each point in the distribution
- Note that the cumulative percentage is associated with the upper real limit of its interval.
What’s the 81st percentile (careful remember the X values are not points on a scale, but rather intervals)?
What is the percentile rank for 70.5?

18. Add following info to your in group table
cumf
cum%

19. Interval Real limits f cf c% Midpoint 92-95 91.5-95.5 4 64 100 93.5
88-91 5 60 94
89.5
84-87 6 55 86
85.5
80-83 1 49 77
81.5
76-79 9 48 75
77.5
72-75 39 61
73.5
68-71 8 34 53
69.5
64-67 7 26 41
65.5
60-63 19 30
61.5
56-59 11 17
57.5
52-55
53.5
How about the percentile rank for X = 59.5?
What’s the percentile rank for X = 91.5?
What’s the 86th percentile?
How about the 61st percentile?

20. Obtaining PR or Interpolation When values don’t appear in the table
Class grades f cum f cum prop c %
Some important symbols:
*Cumfll = cf at lower real limit of X
*c% = cf/ N (100%) *X = score
*Xll = score at lower real limit of X *i = interval width
*fi = # of cases in X’s group *N = total # scores

21. Obtaining PR or Interpolation
Class grades f cum f cum prop cum %
What is the PR of 88? Getting PR from score (X).
PR = cumfll + (( X - Xll) / i) (fi) N
X = 88 i = ( ) = 10
cumfll = 22 Xll = 80.5
N = fi = 4
x 100
78.13 %

22. Obtaining PR or Interpolation
Class grades f cum f cum prop cum %
What is the PR of 88? Getting PR from score (X).
using interpolation
81-69 = 12
2.5/10 = a/12
a = 3
81-3 = 78%
= 10
88 is 2.5 units down the interval
90 81
88 X
81 69

23. Obtaining the score (X) from PR
Class grades f cum f cum prop cum %
What is the score that corresponds to a PR of 72?
cumf = (PR x N) / 100 = (72 x 32)/ 100 = 23.04
X = Xll + [ i (cumf - cumll )/ fi ] = [ 10 ( )/ 4] = 83.1

24. Obtaining the score (X) from PR
Class grades f cum f cum prop cum %
What is the score that corresponds to a PR of 72?
= 10
9/12 = a/10
a = 7.5
= 83
81-69 = 12
72 is 9 down
90 81
X 72
81 69

25. Interval Real limits f cf c% Midpoint 92-95 91.5-95.5 4 64 100 93.5
88-91 5 60 94
89.5
84-87 6 55 86
85.5
80-83 1 49 77
81.5
76-79 9 48 75
77.5
72-75 39 61
73.5
68-71 8 34 53
69.5
64-67 7 26 41
65.5
60-63 19 30
61.5
56-59 11 17
57.5
52-55
53.5
How about the percentile rank for X = 73?
What score is at the 88th percentile?

26. Graphs Visual methods to display data: Basics:
Figure: pictorial; photo; drawing
Table: organized numerical info
Graph: pictorial; axes, #s, etc.
Basics:
X-axis or abscissa: horizontal line in the graph; IV
Y-axis or ordinate: vertical line in the graph: DV
Always label axes [graph’s height should be roughly 2/3 to 3/4 the length (see Box 2.1 pg. 49)]
Y starts at 0; continuous, no breaks
X can change start; break; can be discrete

27. Bar Graphs
Bar graph = nominal or ordinal data (usually nonnumerical values)
Each bar = category
Height = frequency
Bars do NOT touch
If uses ordinal data order must be preserved
Can be vertical or horizontal

28. Histogram Histogram = interval and ratio data
Same rules as bar graph EXCEPT bars touch
Usually for discrete data
Width of the bar extends to the real limits of the score

29. Frequency Distribution Polygon or Line Graph
Line graph = interval, ratio data
Usually used for continuous data
A dot is centered above each score
Can also use this type of graph for relative frequencies (proportions) when there is a large amount of data. In this case each dot would be placed at the midpoint of the range 56 57 58 59 60

30. Cumulative Frequency Graph
Cumulative Frequency Graph = can be a bar, histogram, or line.
Uses the proportion or percentage
Line graph version is typically s-shaped or ogive
Always increases

31. Stem and Leaf Displays Data Set 54 81 82 61 97 83 74 67 86 80 68 87
75 81

32. Advantages to Stem and Leaf
Easy to construct
Allows you to identify each and every individual score (frequency distribution just tells you the frequency)
Both a picture and listing of scores (if you turn the stem and leaf display on its side it looks like a histogram)
* Caveat - just seen as a preliminary means for organizing data

33. Distributions 3 characteristics that describe a distribution
Shape
Central tendency: center of distribution
Variability: spread of scores
Shape: technically shape is defined by an equation that prescribes the exact relationship between each X and Y value on the graph

34. Characteristics of Distributions defined by shape
Skewness (Sk): measure of balance in a distribution
Evenly balanced distributions have no Sk; they are normal or symmetrical
Positive Sk (Sk+): tail trails to the right (positive dir.)
Negative Sk (Sk-): tail trails to the left (negative dir.)
Kurtosis (Ku): measure of how peaked a distribution is
Platykurtic: relatively flat
Leptokurtic: relatively peaked
Mesokurtic: neither flat nor peaked

35. Homework - Chapter 2
1, 2, 4-6, 7, 9, 11, 13, 17, 20, 21, 23, 26
For problems 20, 21, and 23 use either the method of finding PR from X and X from PR that I taught today or the method of interpolation in the book. Your choice!