Displaying the Order in a Group of Numbers Using Tables and Graphs Chapter 1

Frequency Given a set of numbers, how can we make sense of them? Frequency: # of scores with a particular value Five students reported that their level of happiness when taking a statistics exam was a 2 on a (0–10) scale The frequency for a rating of 2 would be 5. Frequency table A table displaying the pattern of frequencies over different values

Steps: Making a Frequency (f) Table 1. Make a list of each possible value, from highest to lowest (N = 29) 2. Go one by one through the scores, making a mark for each score next to its value on the list 3. Make a table showing how many times each value on your list was used 4. Figure the percentage of scores for each value

Frequency Table Step 1 Step 1: Make a list down the page of each possible value, from lowest to highest. Your research study used a happiness scale that ranges from 0 (not at all happy) to 10 (extremely happy) Happiness Rating 1 2 3 4 5 6 7 8 9 10

Frequency Table Step 2 Step 2: Go one by one through the scores, making a mark for each next to its value on the list. Your study resulted in the following scores: 8,2,3,2,9,1,5,6,1,9,4,4,2,3,3,5,4,7,5,3 Happiness Rating Frequency Tally 1 II 2 III 3 IIII 4 5 6 I 7 8 9 10

Frequency Table Step 3 Step 3: Make a table showing how many times each value on your list was used. Happiness Rating Frequency Tally Frequency 1 II 2 III 3 IIII 4 5 6 I 7 8 9 10

Completed Frequency Table Step 4: Figure the percentages of scores for each value Take the frequency of the value, divide it by the total number of scores, and multiply by 100 (N = 20): 2/20= .1 X 100 = 10; 3/20 = .15 X 100 = 15 Happiness Rating Frequency Percent 1 2 10 3 15 4 20 5 6 7 8 9

Frequency Tables for Nominal Variables Follow the same four steps that you would for a numeric variable. Remember that the values in which you are interested are names or categories rather than numbers. Major Frequency Percent Psychology 5 25 Sociology 8 40 Anthropology 3 15 Political Science 4 20

Grouped frequency table Range of scores in each of several equally sized intervals Combines values into intervals Make Large Distributions more Comprehensible Grouped Frequency Table (Interval = 10)

Grouped Frequency Table A frequency table: uses intervals of values Lists the number of participants for each interval of values If list of possible values range from 0–10 a possible set of intervals: 0–1 2–3 4–5 6–7 8–9 10–11

Rules: Constructing Group Frequency Table There should be from 5-15 Intervals All Intervals should have ame size Low end of each interval is always a multiple of the interval size Make intervals a round number (2, 3,5) Make it meaningful to you and the reader.

Histogram Graph of the information on a frequency table The height of each bar is the frequency of each value in the frequency table.

Y Axis: Ordinate X Axis: Abscissa

Frequency Graphs:Histogram Midpoint = Middle of Interval =(Lower Value of Interval + Lower Value of Next Interval Up)/2 Example: 0+5/ 2 = 2.5 5+ 10 = 15/2 = 7.5

Frequency Distributions Show the pattern of frequencies over the various values (how the frequencies are spread out). unimodal distribution a histogram with one very high area bimodal distribution a distribution with two fairly equal high points multimodal distribution a distribution with two or more high points rectangular distribution when all values have approximately the same frequency

Shapes of Frequency Distributions Unimodal Having one peak or one high point Bimodal Having two peaks Multimodal Having two or more peaks Rectangular Having no peaks

Symmetrical and Skewed Distributions In the social and behavioral sciences, most scores are symmetrically distributed. They have approximately the same number of scores on both sides of the distribution. Skewed distributions are distributions where the scores pile up on one side of the middle. On the other side of the middle, the scores are spread out.

Symmetrical vs. Skewed Frequency Symmetrical distribution (a) Approximately equal numbers of observations above and below the middle Skewed distribution (b-c) One side is more spread out than the other, like a tail

Skewed Distributions Characterized by the side of the distribution where scores are more spread out (tail) negatively skewed distribution tail is to the left positively skewed distribution tail is to the right Copyright © 2011 by Pearson Education, Inc. All rights reserved.

Skewed Frequency Distributions Skewed right (b) Fewer scores right of the peak Positively skewed Can be caused by a floor effect Skewed left (c) Fewer scores left of the peak Negatively skewed Can be caused by a ceiling effect

Copyright © 2011 by Pearson Education, Inc. All rights reserved.

Ceiling and Floor Effects Ceiling effects Occur when scores can go no higher than an upper limit and “pile up” at the top e.g., scores on an easy exam, as shown on the right Causes negative skew Floor effects Occur when scores can go no lower than a lower limit and pile up at the bottom e.g., household income Causes positive skew

Normal, Heavy-Tailed, and Light-Tailed Distributions Normal Curve bell-shaped, unimodal, and symmetrical Heavy-Tailed Distribution There are many scores in the tails (the tails are thick). Light-Tailed Distribution There are few scores in the tails (the tails are thin).

Issues with Normal Distributions Skewness Kurtosis

Kurtosis Degree to which tails of the distribution are Light vs. Heavy! Bell-Shaped/Symmetrical Light-Tailed Heavy-Tailed Mesokurtic Platykurtic Leptokurtic Middle Flat Narrow

Frequency Tables and Histograms in Research Articles Research articles use histograms and frequency tables to describe the data prior to discussing more complex statistical procedures. However, frequency tables and histograms are not often found in published research. It is particularly important to describe the data distributions when the distribution of values is not a normal distribution.

Key Points Descriptive statistics are used to describe and summarize a group of numbers from a research study. A value is a number or category; a variable is a characteristic that can have different values; a score is a particular person’s value on the variable. Some numeric variables are rank-ordered and some variables are names or categories and not numbers. A frequency table organizes the scores into a table that lists each possible value from lowest to highest along with the frequency of each value. A grouped frequency table is used when there are many different values. Intervals are given for a range of values. A histogram visually displays the information in a frequency table. The general shape of a histogram can be unimodal, bimodal, multimodal, or rectangular, and the distribution can be symmetrical, skewed to the right, or skewed to the left. Frequency tables, when used in research articles, are used to summarize the characteristics of study participants. Histograms almost never appear in articles, but the shapes of the distribution are sometimes described in words.