COMM 301: Empirical Research in CommunicationDescriptive Statistics: Measures of VariabilityDescriptive Statistics: Describing Data with Numbers – PART 2Measures of VariabilityVariability refers to how dispersed are the data points in a distribution, and how similar or different each data point is from the other data points.There are three common measures of variability: range, variance, and standard deviation.RangeWhat is it?The range is the distance or difference between the lowest score and the highest score.How to find the range?For any given set of scores, subtract the lowest score from the highest score.For example, a data set has 10 data points (on a 9-point scale): 5, 4, 5, 5, 4, 5, 6, 4, 1, 9The range would be: 9 - 1 = 8. When is it used?Type of question answered- What does this set of scores look like?- How variable are the scores in the distribution?Type of data- Variables: One (1) continuous variable- Measurement levels: Interval, ratioWhat do you need to know?All the above, plus- the range is affected by extreme scores;- recognition of SPSS output. How to report descriptive statistics?See the following example.1COMM 301: Empirical Research in CommunicationDescriptive Statistics: Measures of VariabilityVariance and Standard DeviationWhat are they?The variance is a measure of the amount that a set of data varies about its mean. Variance is akey concept, and it forms the heart of all statistics. Standard deviation is the undoing of the squaring that we did to find the variance. The standard deviation therefore is really a sort of "average distance" of each point from the mean..  very important concept in normal distribution.The standard deviation is the square root of the varianceHow to find the variance and standard deviation?For a given population or universe of scores, the formula for variance is [(first score - mean score)2 + (second score - mean score) 2 + … + (last score - mean score) 2] __________________________________________________________________________ number of scoresIf the set of scores is a sample, then the formula for variance is[(first score - mean score)2 + (second score - mean score) 2 + … + (last score - mean score) 2] __________________________________________________________________________number of scores - 1The standard deviation is the square root of the variance.When are they used?Type of question answered- What does this set of scores look like?- How variable are the scores in the distribution?Type of data- Variables: One (1) continuous variable- Measurement levels: Interval, ratioWhat do you need to know?All the above, plus- variance and standard deviation are meaningful only for continuous variables, measured with interval or ratio scales;- recognition of SPSS output. 2COMM 301: Empirical Research in CommunicationDescriptive Statistics: Measures of VariabilityHow to report descriptive statistics?See the following example.Procedures in SPSSAnalyze > Descriptive Statistics > Frequencies …Select variable.Select the appropriate chart.Click “Statistics”.Check “Mean”, “Median” “Mode”.Check “Minimum”, “Maximum”, “Std. deviation”, “Variance”, “Range”.Click “Paste.”Go to the syntax file. Highlight the appropriate section, and click ►.3COMM 301: Empirical Research in CommunicationDescriptive Statistics: Measures of VariabilityResult (example from lect13_4 data set ( also show bar chart example)StatisticsEXAMSCOR N Valid 20Missing 0Mean 79.2000Median 80.0000Mode 80.00Std. Deviation 7.34560Variance 53.95789Skewness .811Std. Error of Skewness .512Kurtosis 2.498Std. Error of Kurtosis .992Range 34.00Minimum 66.00Maximum 100.00EXAMSCOR100.095.090.085.080.075.070.065.0EXAMSCORFrequency86420Std. Dev = 7.35 Mean = 79.2N = 20.00ReportIn this group of 20 students, the mean score was 79.20 (standard deviation = 7.35). The range was 34, with the highest score = 100, and the lowest score = 66.4COMM 301: Empirical Research in CommunicationDescriptive Statistics: Measures of VariabilityDescribing Data with PicturesThere are several ways to describe data. One of way is to use pictures. These techniques include frequency distributions, histograms and bar graphs. We will focus on histograms and bar graphs.HistogramsWhat are they?A histogram shows in a picture form how many times a given score appears in a data set. There are two main axes on the histogram. The horizontal axis (the X axis) is where the scores are represented. Typically the scores are grouped into score intervals. Each score interval is represented by one rectangular bar, and the mid-point of the score interval is highlighted. The rectangular bars touch each other, because the data is continuous. The vertical axis (the Y axis) indicates the frequency of those scores occurring.A tall bar indicates a high frequency of occurrence, meaning that the score occurs many times. A short bar indicates a low frequency of occurrence, meaning that the score occurs fewtimes.Example90.087.585.082.580.077.575.072.570.067.5Raw score in courseFrequency1086420Std. Dev = 5.20Mean = 84.1N = 20.00ReportThe raw scores of the 20 students ranged from 66.89 to 91.01, with a mean raw score of 84.1. A histogram depicting the continuum of scores reveals that the peak score interval is between 83.75 and 88.75. The distribution of the scores is negatively skewed, with most of the scores on the higher ranges.5COMM 301: Empirical Research in CommunicationDescriptive Statistics: Measures of VariabilityBar ChartsWhat are they?A bar chart shows in a picture form how many times a given score appears in a data set. Bar charts are almost identical to histograms, except that they deal with categorical data, rather than continuous data. The horizontal axis (the X axis) is where the categories are represented. Each category is represented by one bar. The vertical axis (the Y axis) indicates the frequency of membership in a category. A tall bar indicates a high frequency of membership, meaning that the category has many members. A short bar indicates a low frequency of membership, meaning that the category has few members.ExampleRaw grade for courseRaw grade for courseDCC-plusB minusBB-plusA minusFrequency121086420ReportThe above bar charts show the distribution of the raw grades earned by the students. One student earned a grade of A-; four students earned a grade of B+; ten students earned a grade of B; two students earned a grade of B-; one student earned a grade of C+; one