1. Lecture 2 PY 427 Statistics 1 Fall 2006 Kin Ching Kong, Ph.D
Chicago School of Professional Psychology
Lecture 2
Kin Ching Kong, Ph.D

2. Agenda Frequency Distributions Central Tendency Tables Graphs
Relative Frequencies & Population Distributions
The Shape of a Frequency Distribution
Central Tendency
Mean
Median
Mode
Selecting a Measure of Central Tendency
Central Tendency & Shapes of Distributions

3. Frequency Distributions
Frequency Distribution: an organized tabulation of the number of individual scores located in each category on the scale of measurement.
Two Elements in Frequency Distributions:
The set of categories that made up the measurement scale.
The frequency, or number of scores in each category.
Frequency distribution: allows one to see the entire set of scores at a glance. Can see a score relative to all other scores. Frequency Distributions can be presented as a table or a graph.

4. Frequency Distribution Tables
Two Basic Columns:
X lists all the measurement categories
f lists the frequency, or number of time that category (X value) occur in the data set.

5. An Example
On a homework assignment with 5 possible points, the scores are:
3, 5, 4, 4, 5, 4, 5, 3, 5, 2, 5, 3, 3, 4, 5, 5, 4, 2, 5, 4
S f = N X f 5 4 3 2 1 8 6

6. Frequency Distribution Table (p & %)
Two common additional columns to frequency distribution tables: Proportions & Percentages.
proportion = p = f/N
Percentage = p(100) X f
p= f/N
%= p(100) 5 4 3 2 1 8 6
.40
.30
.20
.10
40%
30%
20%
10% 0%

7. Grouped Frequency Distribution Tables
Grouped Frequency Distribution Tables: when the range of scores in a data set is large, we group the scores into intervals, called class intervals.
Example
E.g. 2.3 in your book, N = 25 exam scores ranging from 53 to 94:
82, 75, 88, 93, 53, 84, 87, 58, 72, 94, 69, 84, 61, 91, 64, 87, 84, 70, 76, 89, 75, 80, 73, 78, 60
42 rows will be needed to represent the original measurement categories
Rule1: about 10 class intervals Rule 2: the width of each interval should be a simple number Rule 3: Each class interval begin with a multiple of the width; Rule 4: All interval should be the same width.

8. Frequency Distribution Graphs
Histograms (for Interval or Ratio Data):
X-axis: list the categories of measurement
Y-axis: list the frequencies (or %)
Draw a Bar above each X value such that:
The height of the bar corresponds to the frequency for that category.
The width of the bar extends to the real limits of each category
An Example
An Example of Grouped Data
Depending on how a variable is used in research, it is either called an independent variables or a dependent variable. Both continuous and discrete variables can be independent or dependent variables.

9. Frequency Distribution Graphs
Polygons (for Interval or Ratio Data):
X-axis: list the categories of measurement
Y-axis: list the frequencies
Center a dot above each X value such that:
The height of the dot corresponds to the frequency for that category.
Connect the dots
Extend the line to cross the X-axis one category below and one category above the two end categories.
An Example
An Example of Grouped Data
Depending on how a variable is used in research, it is either called an independent variables or a dependent variable. Both continuous and discrete variables can be independent or dependent variables.

10. Frequency Distribution Graphs
Bar Graphs (for Nominal or Ordinal Data):
Essentially the same as Histograms except a space is left between bars
An Example

11. Relative Frequencies & Population Distributions
For Populations
Relative Frequencies
Smooth Curves
An Example

12. Shape of a Frequency Distribution
Symmetrical
Skewed
Positively skew (tail on the right)
Negatively skew (tail on the left)

13. Three Measures of Central Tendency:
Central Tendency: a single value that identify the center of a distribution, or is the most typical or representative of the entire distribution.
Three Measures of Central Tendency:
The Mean
The Median
The Mode

14. The Mean The Mean = Arithmetic Average Characteristics of the Mean
Population mean: m = S X/N
Sample mean: M = S X/n
Characteristics of the Mean
Change, add, remove a score will change the Mean (expect when added/removed score = the mean)
Add or subtract a constant from each score, the mean will be changed by the same constant
Multiply or divide each score by a constant, the mean will be multiplied or divided by the same amount.

15. The Median
The Median: the score that divides a distribution exactly in half. 50% of the scores are >= median, 50% <=Median
To find the Median:
With an odd number of scores: find the middle score
With an even number of scores: find the average of the middle two scores.

16. The Mode: the score or category that has the greatest frequency.
The Mode can be used with any scales of measurement, including nominal
A distribution can have multiple modes
The mode(s) identify the location of the peak(s) in the frequency distribution.

17. Selecting a Measure of Central Tendency
Scale of Measurement:
The Mean: interval or ratio data
The Median: ordinal, interval or ratio Data
The Mode: all scales of measurement
The mean is usually the preferred measure for interval or ratio data except for:
Extreme scores or skewed distributions:
The mean may not be a good measure of Central Tendency
Median is often preferred. (e.g. median income)
Example
Undetermined values
Mean cannot be computed, but Median can be obtained.
Discrete variables
Mean often produce fractional value that cannot exist in real life. Mode makes more sense. (e.g. the modal family has 2 kids)
The mean in generally is the preferred measure of central tendency.

18. Central Tendency & Shapes of Distributions
Symmetrical Distributions
Mean and Median will be the same.
When there is only one mode, it will the same as the mean and median
Figure 3.11
Skewed Distributions:
Positively Skewed: mode, median, mean
Negatively Skewed: mean, median, mode
Figure 3.12