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

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

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.

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.

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

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%

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.

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.

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.

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

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

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

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

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.

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.

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.

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.

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