Let s take a closer look at the standard deviation.

Transcription

1 For this illustration, we have selected 2 random samples of 500 respondents each from the voter.xls file we have been using in class. We recorded the ages of each of the 500 respondents and computed the descriptive statistics summarized in the following table. Sample 1 Sample 2 Count (n) Min Q Median Q IR Max Range Mean StDev CV Based on these statistics, we could conclude that: 1) Sample 1 is slightly older that Sample 2- note the medians and means for both samples; 2) Both samples have the same range of ages- 67 years; 3) Sample 1 is slightly more variable in age than Sample 2- note the interquartile ranges, standard deviations, and coefficients of variation for both samples. We can present a more detailed comparison of the differences in central tendcncy and variability between these 2 samples using a graphic display. We have done this on the following slide.

2 Range: 100% of the distribution lies between these extreme numbers. Interquartile range: the middle 50% of the distribution lies between these numbers. Sample 1 ( ) ( ) [ ] Min Mean -1s Q1 Mean Q3 Mean +1s Max [ Range ] [Mean 1s Mean + 1s] [ Interquartile Range ] Sample 2 ( ) ( ) [ ] Min Mean -1s Q1 Mean Q3 Mean+1s Max Let s take a closer look at the standard deviation.

3 Let s start with Sample 1. In this example, the mean of Sample 1 = and the standard deviation = We can subtract the standard deviation from the mean: = Graphing the lower half of the distribution- that is, the part of the distribution from the Minimum age (22) to the mean age (48), we see: Youngest Approximately 34% of the people in this Mean person [Mean-1s] <- sample are between these 2 points. - > age We can also add the standard deviation to the mean. For Sample 1, this gives = Graphing the upper half of the distribution, we see: Mean Approximately 34% of the people in this Oldest Age <- sample are between these 2 points. -> [Mean +1s] person Putting these graphs of the 2 halves of the distribution gives us the graph from the preceding slide. [ ] Min Mean -1s Q1 Mean Q3 Mean +1s Max

4 Quickly applying the same approach to the data from Sample 2, where the mean = and standard deviation = Youngest Approximately 34% of the people in this Mean person [Mean-1s] <- sample are between these 2 points. - > age And Mean Approximately 34% of the people in this Oldest Age <- sample are between these 2 points. -> [Mean +1s] person Again, combining these graphs results in the second graph from a preceding slide. ( ) ( ) [ ] Min Mean -1s Q1 Mean Q3 Mean+1s Max

5 Note that we keep stating that approximately 34% or 68% of the distribution lines between the mean and + 1s. In most samples, the percentage will not be exactly 68, but this is still a good rule of thumb to use in interpreting what the standard deviation tells us about the variability or dispersion in a distribution of data. To conclude this presentation with a verbal statement of the interpretation of these measures of variability or dispersion. 1) The range [highest number lowest number] indicates the variability or dispersion between The 2 extreme numbers in the data set; it does not indicate anything about the variability of the numbers inside the data set. 2) The interquartile range [third quartile first quartile] indicates the variability or dispersion of the middle 50% of the numbers in the data set. Note that, while it does provide more information than the range, it does not take into account every number in the data set. 3) The standard deviation indicates the overall variability or dispersion of the numbers in the data set. By overall variability or dispersion, we mean it takes into account every number in the data set. It thus provides the most powerful measure of variability or dispersion. It can also be used in the calculation of the coefficient of variation. By adding 1 standard deviation to the mean, we can find the part of the data set containing approximately 34% of the data; by subtracting 1 standard deviation from the mean, we can find the part containing another 34% of the data. Between these 2 points, we find approximately 68% of the data in the set. And, for any of these measures, the smaller the measure, the less variability or dispersion in the data set.