1. Smith/Davis (c) 2005 Prentice Hall Chapter Four Basic Statistical Concepts, Frequency Tables, Graphs, Frequency Distributions, and Measures of Central Tendencies PowerPoint Presentation created by Dr. Susan R. Burns Morningside College

2. Smith/Davis (c) 2005 Prentice Hall Two Branches of Statistics Statistics is a branch of mathematics that involves the collection, analysis, and interpretation of data. The two main branches of statistics assist your decisions in different ways: Descriptive Statistics – are used when you want to summarize a set or distribution of numbers in order to communicate their essential characteristics. (e.g., measure of central tendency). Inferential Statistics – are used to analyze data after you have conducted an experiment to determine whether your independent variable had a significant effect.

3. Smith/Davis (c) 2005 Prentice Hall Measurement Measurement is defined as the assignment of symbols to events according to a set of rules. Scales of Measurement – are the particular set of rules in assigning a symbol to the event in question. Nominal Scale – is a simple classification system (e.g., categorizing types of ice-cream). Ordinal Scale – is used when events in question can be rank ordered. With the ordinal scale, the intervals separating the ranks do not have to be comparable. Interval Scale – is used when the events in question can be rank ordered and equal interval s separate adjacent events. Ratio Scale – takes the interval scale one step further by permitting the rank order of scores with the assumption of equal intervals between the ranks, but it also assumes the presence of a true zero point.

4. Smith/Davis (c) 2005 Prentice Hall Scales of Measurement

5. Smith/Davis (c) 2005 Prentice Hall Frequency Distributions Frequency Distributions – show how often each score occurs in your research. – Some easy steps to follow that will help you construct a frequency distribution are: In one column make a list of the categories for which you have frequencies. If you categories are ordinal, interval, or ratio arrange them in order from highest to lowest. Create a second column to the right of the first column. Label this column “tally.” Create a third column to the right of the second. Label this column “frequencies.” Convert the tallies to the frequencies and record them in the frequency column. Calculate the percentage of occurrence for each category by dividing the frequency for each category by the total number of scores and then multiply by 100. This figure appears in the final column.

6. Smith/Davis (c) 2005 Prentice Hall Frequency Distributions

7. Smith/Davis (c) 2005 Prentice Hall Frequency Distributions Frequency Distributions – show how often each score occurs in your research. Grouped Frequency Distributions – if you can order your categories numerically from lowest to highest, then you may want to group your frequencies into intervals.

8. Smith/Davis (c) 2005 Prentice Hall Frequency Distributions

9. Smith/Davis (c) 2005 Prentice Hall Graphing Your Results There are several types of graphs from which the researcher can choose. Your choice of graphs will be determined by which one depicts your results most effectively and by the scale of measurement you used.

10. Smith/Davis (c) 2005 Prentice Hall Types of Graphs Pie Chart – depicts the percentage represented by each alternative as a slice of a circular pie; the larger the slice, the greater the percentage.

11. Smith/Davis (c) 2005 Prentice Hall Types of Graphs Bar Graph – presents data in terms of frequencies per category. You will construct a bar graph when you are using nominal (or qualitative) categories that cannot be numerically ordered from lowest to highest.

12. Smith/Davis (c) 2005 Prentice Hall Types of Graphs Histogram – represents quantitative data in terms of frequencies.

13. Smith/Davis (c) 2005 Prentice Hall Types of Graphs Frequency Polygon – like the histogram, displays the frequency of each number or score. The only differences between these two graphs are the use of bars in the histogram and the use of connected dots in the frequency polygon.

14. Smith/Davis (c) 2005 Prentice Hall Types of Graphs Line Graph – in line graphs, there are two axes or dimensions that must be discussed. The vertical (Y axis) is known as the ordinate; the horizontal (X axis) is known as the abscissa. One of your variables is plotted on the ordinate and the other is plotted on the abscissa. A good guideline is to plot the variable that has the greatest number of levels on the abscissa, and thus reducing the number of lines that will appear on your graph.

15. Smith/Davis (c) 2005 Prentice Hall Types of Graphs Line Graph – How tall should the Y axis be? How long should the X axis be? A good rule of thumb is for the U axis to be approximately two thirds as tall as the X axis is long. Other configurations will give a distorted picture of the data.

16. Smith/Davis (c) 2005 Prentice Hall Distributions and Their Shapes Statisticians use several specific terms to describe the different shapes these distributions can assume. – Unimodal Distributions have one prominent category or high point. – Bimodal Distributions have two prominent categories or high points. – Multimodal Distributions have several prominent categories or high points.

17. Smith/Davis (c) 2005 Prentice Hall Distributions and Their Shapes Statisticians use several specific terms to describe the different shapes these distributions can assume. – Rectangular Distributions occur when there is no prominent category or high point, when frequencies are the same in all the same.

18. Smith/Davis (c) 2005 Prentice Hall Measures of Central Tendency Measures of central tendency tell us about the typical score in a distribution. There are three measures of central tendency: – Mode – is the number or event that occurs most frequently in a distribution. – Median – is the number or score that divides the distribution into equal halves (i.e., the median is the 50th percentile). To be able to calculate the median, you must first rank order the scores. – Mean – the mean is defined as the arithmetic average. To find the mean, you add up all the scores in the distribution and then divide by the number of scores that are added.

19. Smith/Davis (c) 2005 Prentice Hall Measures of Central Tendency Choosing a Measure of Central Tendency: – If you want to know which score occurred most often, then the mode is the choice. – The median is a better choice to serve as the representative score because it takes into account all the data in the distribution. However, it treats all scores alike; differences in magnitude are not taken into account. – When the mean is calculated, the value of each number is taken into account. When the scores in your distribution tend to cluster in one of the tails (i.e., a cluster of high or low scores) the distribution is skewed (i.e., a nonsymmetrical distribution). In these instances, the median may be more appropriate.

20. Smith/Davis (c) 2005 Prentice Hall Measures of Central Tendency When the scores in your distribution tend to cluster in one of the tails (i.e., a cluster of high scores or a cluster of low scores) the distribution is skewed. – Positively Skewed Distributions – occur when there is cluster of lower scores, the smaller, more spread-out tail will be on the right (i.e., fewer high scores). Figure A – Negatively Skewed Distributions – occur when there is a cluster of higher scores, the smaller more spread out tail will be on the left (i.e., fewer small scores). Figure B