Week4

Descriptive Statistics : 

Descriptive Statistics Soleman Abu-Bader, PhD

Frequency Distribution : 

Dr. Abu-Bader Quantitative Methods 2 Frequency Distribution It is the first step in data analysis. It is an arrangement of values that shows the number of times a given score or group of scores occurs. It helps us to know “how the research sample or population broke”. It gives the frequency of each value for each variable. Score = X Frequency – f

Arranging Data: Array : 

Dr. Abu-Bader Quantitative Methods 3 Arranging Data: Array Data can be arranged in different ways. First, arrange the scores in an array. In this case, scores are arranged from the lowest value to the highest value. Next, count the number of times each score, X, occurs. This is the frequency, f.

Arranging Data: Example : 

Dr. Abu-Bader Quantitative Methods 4 Arranging Data: Example RAW DATA 86 80 82 86 88 90 88 90 94 92 84 88 84 88 86 92 88 96 90 88 ARRAY80 82 84 84 86 86 86 88 88 88 88 88 88 90 90 90 92 92 94 96

Types of Frequency Distributions : 

Dr. Abu-Bader Quantitative Methods 5 Types of Frequency Distributions Absolute frequency distribution (f) Cumulative frequency distribution (cf) Percentage distribution (%) Cumulative percentage distribution (c%)

Frequency Distribution Table: Example : 

Dr. Abu-Bader Quantitative Methods 6 Frequency Distribution Table: Example

Working with Large Data Files : 

Dr. Abu-Bader Quantitative Methods 7 Working with Large Data Files

Frequency Distribution: Final Scores : 

Dr. Abu-Bader Quantitative Methods 8 Frequency Distribution: Final Scores

Class Interval : 

Dr. Abu-Bader Quantitative Methods 9 Class Interval Reduce the number of scores by grouping several scores into an interval of scores (class interval). These class intervals must have equal width. When possible, the width should be an odd number; e.g., 3, 5, 7, etc. This makes it easier to determine the midpoint of a class interval.

Final Exam Scores In Class Intervals : 

Dr. Abu-Bader Quantitative Methods 10 Final Exam Scores In Class Intervals

Exact Limits of Class Interval : 

Dr. Abu-Bader Quantitative Methods 11 Exact Limits of Class Interval When class intervals are used, the scores are considered discrete values. In our example, final scores are continuous data and thus must include all possible outcomes.

Final Exam Scores: Exact Limits : 

Dr. Abu-Bader Quantitative Methods 12 Final Exam Scores: Exact Limits

Graphs : 

Dr. Abu-Bader Quantitative Methods 13 Graphs Helps us to “eye ball” the information collected

Graphing Data: Constructing a Graph : 

Dr. Abu-Bader Quantitative Methods 14 Graphing Data: Constructing a Graph Draw two coordinate axes (X axis, Y axis) Plot the data on the graph

Bar Graphs : 

Dr. Abu-Bader Quantitative Methods 15 Bar Graphs It is used to represent the distribution of value categories for nominal level variables. Bars of equal width are drawn so that they do not touch. This suggests that the categories are mutually exclusive. The y axis represents the frequency and x axis represents the categories.

Bar Graph: Example : 

Dr. Abu-Bader Quantitative Methods 16 Bar Graph: Example

Histogram : 

Dr. Abu-Bader Quantitative Methods 17 Histogram It is used to represent the distribution of scores for interval or higher level variables. A histogram, like a bar graph, uses the height of a bar to display frequency (y axis) of a given variable. Unlike a bar graph, the bars in histogram touch each other representing continuous data.

Histogram: Example : 

Dr. Abu-Bader Quantitative Methods 18 Histogram: Example

Stem-and-Leaf Plot : 

Dr. Abu-Bader Quantitative Methods 19 Stem-and-Leaf Plot In stem-and-leaf plots, each value is presented. The first step in developing stem-and-leaf plot is to determine the stem. Example: 20 21 21 22 22 30 31 32 33 33 33 34 34 40 45 45 46 47 52 52 53 54 First, we will use the 10s place (20, 30, 40, 50) to represent the STEM. Second, we will use the Ones place (0 to 9) to represent the leaf.

Stem-and-Leaf Plot: Example : 

Dr. Abu-Bader Quantitative Methods 20 Stem-and-Leaf Plot: Example Data: 20 21 21 22 22 30 31 32 33 33 33 34 34 40 45 45 46 47 52 52 53 54

Frequency Polygon : 

Dr. Abu-Bader Quantitative Methods 21 Frequency Polygon Here we assume that the midpoint scores represent the class intervals. For each interval, we plot the frequency of scores at the midpoint. Next, connect these midpoints.

Frequency Polygon - Example : 

Dr. Abu-Bader Quantitative Methods 22 Frequency Polygon - Example

Cumulative Frequency Distribution : 

Dr. Abu-Bader Quantitative Methods 23 Cumulative Frequency Distribution

Ogive Graph : 

Dr. Abu-Bader Quantitative Methods 24 Ogive Graph It represents the cumulative frequency.

Percentiles : 

Dr. Abu-Bader Quantitative Methods 25 Percentiles The formula to compute percentiles is as follows: ll=exact lower limit n=total # of scores p=proportion corresponding to desired percentile cf=cumulative frequency of scores below the interval containing the percentile point fi=frequency of scores in this interval w=width of interval

Percentiles - Example : 

Dr. Abu-Bader Quantitative Methods 26 Percentiles - Example What is the 75th Percentile in the previous data?

Percentile Rank : 

Dr. Abu-Bader Quantitative Methods 27 Percentile Rank This is the percent of scores less than or equal to that score. The PR63 is the percent of scores in the distribution that fall at or below 63. The exact PR is:

Percentile Rank - Example : 

Dr. Abu-Bader Quantitative Methods 28 Percentile Rank - Example What is the PR for a score of 61?

Measures of Central Tendency : 

Measures of Central Tendency Mode Median Mean

Mode : 

Dr. Abu-Bader Quantitative Methods 30 Mode It is the simplest index of central tendency. It is the most frequent score in the distribution. It is simply determined by counting the data; the number of times each value occurs. The value that occurs most frequently is the mode. When there is only one mode, the distribution is called unimodal. When there are two modes, it is called bimodal. When it has three or more modes, it is called multimodal.

What is the Mode? : 

Dr. Abu-Bader Quantitative Methods 31 What is the Mode? Example Data:80 82 84 84 86 86 8688 88 88 88 88 88 9090 90 92 92 94 96 Mode = 88

Median : 

Dr. Abu-Bader Quantitative Methods 32 Median It is the point on a scale of measurement below which 50% of scores fall. To determine the median, first arrange the data from the lowest value to the highest value. In other words, the median divides an array of values into two equal halves. If the distribution has an odd number of cases, the median is the middle number If the distribution has an even number of cases, the median is the sum of the two middle scores divided by 2.

What is the Median? : 

Dr. Abu-Bader Quantitative Methods 33 What is the Median? 80 82 84 84 86 86 86 88 88 88 88 88 88 90 90 90 92 92 94 96 Median = (88+88)/2 = 88

Mean : 

Dr. Abu-Bader Quantitative Methods 34 Mean The mean is the arithmetic average of the scores in a distribution. The symbol for the population mean is μ, and the sample mean is

Formula of the Mean : 

Dr. Abu-Bader Quantitative Methods 35 Formula of the Mean Formula: The sum of all score frequencies divided by the total number of cases.

Which Measure to Use? : 

Dr. Abu-Bader Quantitative Methods 36 Which Measure to Use? It is not always easy to decide which measure of central tendency to use. The choice depends on the data Whether or not there are any outlier cases, for example. With nominal data, the mode is the only measure to be used. With ordinal data, the median, in many cases, is appropriate ... sometimes the mean is used. With interval or ratio data, the threemeasures can be used.

Slide 37: 

Dr. Abu-Bader Quantitative Methods 37