Wk 4 Discussion 1 - Measures of Variability due Thurs

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Wk 4 Discussion 1 - Measures of Variability [due Thurs] Due Thursday M easures of variability include the range, the variance, and the standard deviation, and define how far away the data points tend to fall from the center. Write a 250- to 300-word response to the following:  How do you choose which measure of variability to use and what considerations may have an impact on your decision? Include your own experience as well as 2 citations that align with or contradict your comments as sourced from peer-reviewed academic journals, industry publications, books, and/or other sources. Cite your sources according to APA guidelines. If you found information that contradicts your experience, explain why you agree or disagree with the information. Measures of variability are numbers that variability in a distribution of data. There are five measures of variability: the index of qualitative variation, the range, the interquartile range, standard deviation, and the variance. Researchers will often use measures of variability along with measures of central tendency to describe the data in a study (Frankfort-Nachmias et al, 2019). Measures of central tendency reveal the typical value in data; however, measure of variability shows how far away data points fall on the distribution and how far spread out the data points are from the center of the distribution (Tokunaga & , 2018). The index of qualitative variation (IQV) is a partial function of the numbers category and is a measure of variability for nominal variables. The IQV is the ratio of the total number of differences in a distribution to the max number of potential differences in the same distribution. One of the simplest measures of variation is called the range, which is used to measure interval-ration variables. The range is the difference between the maximum score and the minimum score in a distribution. A third measure of variability is the interquartile range (IQR), which measures the ordinal and interval-ratio variables. The interquartile range remedies limitations of the range and is the width of the middle 50% of a distribution and is calculated by taking the difference between lower and upper quartiles. The final two measures of variation are standard deviation and variance, which are closely related as they both decrease or increase related to how close the scores cluster around the mean. The standard deviation and variance are measures of variation for interval-ratio and ordinal variables. Standard deviation is the square root of the variance, and the variance is an average of the squared deviations from the center of the distribution. Unlike the variance, the standard deviation is measured using the same units as the original data (Frankfort-Nachmias et al, 2019). Above the five measures of variation are discussed and defined, however the variables level of measurement guides which measure of variability to use. Nominal variables use IQV to determine the variability in a distribution. An ordinal level of measurement’s research objective will use IQV to measure variability however in doing so it will ignore any rank or order. In addition, an ordinal level of

measurement’s research objective will determine variability in a distribution by using variance and standard deviation as well as measure range of rank ordered categories by using range and IQR. For an interval-ratio level of measurement, the shape of the distribution determines the measure of variability. If the shape of distribution is symmetrical, the interval-ratio’s research objective will use range and IQR for the rough assessment of variation as well as use standard deviation and variance to determine the variability of the distribution. If the shape of distribution is skewed, the interval-ratio will use range and IQR to measure variability (Frankfort-Nachmias et al, 2019). Frankfort-Nachmias, C., Leon-Guerrero, A., & Davis, G. (2019). Social statistics for a diverse society (9th ed.). SAGE Publications. Tokunaga, H. T. . (2018). Fundamental statistics for the social and behavioral sciences (2nd ed.). SAGE. https://uk.sagepub.com/sites/default/files/upm-assets/95832_book_item_95832.pdf

Nominal level: With nominal variables, your choice is restricted to the IQV as a measure of variability. Ordinal level: The choice of measure of variation for ordinal variables is more problematic. The IQV can be used to reflect variability in the distributions of ordinal variables, but because IQV is sensitive to the rank-ordering of values implied in ordinal variables, it loses some information. Another possibility is to use the IQR, interpreting the IQR as the range of rank-ordered values that includes the middle 50% of the observations.11 For example, if the IQR for income categories begins with the category $50,000 to $70,500 and ends with the category $100,000 to $120,500, the IQR can be reported as between $50,000 and $120,500. However, in most instances, social science researchers treat ordinal variables as interval-ratio measures, preferring to calculate variance and standard deviation. Interval-ratio level: For interval-ratio variables, you can choose the variance, standard deviation, the range, or the IQR. Because the range, and to a lesser extent the IQR, is based on only two scores in the distribution (and therefore tends to be sensitive if either of the two points is extreme), the variance and/or standard deviation is usually preferred. However, if a distribution is extremely skewed so that the mean is no longer representative of the central tendency in the distribution, the range and the IQR can be used. The range and the IQR will also be useful when you are reading tables or quickly scanning data to get a rough idea of the extent of dispersion in the distribution. The index of qualitative variation (IQV) is a measure of variability for nominal variables such as race and ethnicity. The index can vary from 0.00 to 1.00. When all the cases in the distribution are in one category, there is no variation (or diversity) and the IQV is 0.00. In contrast, when the cases in the distribution are distributed evenly across the categories, there is maximum variation (or diversity) and the IQV is 1.00. Index of qualitative variation (IQV): A measure of variability for nominal variables. It is based on the ratio of the total number of differences in the distribution to the maximum number of possible differences within the same distribution. It is important to remember that the IQV is partially a function of the number of categories. To summarize, these are the steps we follow to calculate the IQV: Construct a percentage distribution.

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Square the percentages for each category. Sum the squared percentages. Calculate the IQV using formula 4.1. The simplest and most straightforward measure of variation is the range, which measures variation in interval-ratio variables . It is the difference between the highest (maximum) and the lowest (minimum) scores in the distribution: Range = Highest score – Lowest score Range: A measure of variation in interval-ratio variables. It is the difference between the highest (maximum) and the lowest (minimum) scores in the distribution. The range is a is a rather crude measure because it is based on only the lowest and the highest scores. These two scores might be extreme and rather atypical, which might make the range a misleading indicator of the variation in the distribution. To remedy the limitation of the range, we can employ an alternative—the interquartile range. The interquartile range (IQR), a measure of variation for interval-ratio and ordinal variables , is the width of the middle 50% of the distribution. It is defined as the difference between the lower and upper quartiles (Q1 and Q3). Interquartile range (IQR): The width of the middle 50% of the distribution. It is defined as the difference between the lower and upper quartiles (Q1 and Q3). IQR can be calculated for interval-ratio and ordinal data. IQR = Q3 – Q1 Like the range, the IQR is based on only two scores. However, because it is based on intermediate scores, rather than on the extreme scores in the distribution, it avoids some of the instability associated with the range. A graphic device called the box plot can visually present the range, the IQR, the median, the lowest (minimum) score, and the highest (maximum) score. The box plot provides us with a way to visually examine the center, the variation, and the shape of distributions of interval-ratio variables. What can we learn from creating a box plot? We can obtain a visual impression of the following properties: First, the center of the distribution is easily identified by the solid line inside the box. Second, since the box is drawn between the lower and upper quartiles, the IQR is reflected in the height of the box. Similarly, the length of the vertical lines drawn outside the box (on both ends) represents the range of the distribution.8 Both the IQR and the range give us a visual impression of the spread in the

distribution. Finally, the relative position of the box and the position of the median within the box tell us whether the distribution is symmetrical or skewed. A perfectly symmetrical distribution would have the box at the center of the range as well as the median in the center of the box. When the distribution departs from symmetry, the box and/or the median will not be centered; it will be closer to the lower quartile when there are more cases with lower scores or to the upper quartile when there are more cases with higher scores.Box plots are particularly useful for comparing distributions The variance and the standard deviation are two closely related measures of variation that increase or decrease based on how closely the scores cluster around the mean. The variance is the average of the squared deviations from the center (mean) of the distribution, and the standard deviation is the square root of the variance. Both measure variability in interval-ratio and ordinal variables. Variance: A measure of variation for interval-ratio and ordinal variables ; it is the average of the squared deviations from the mean. Standard deviation: A measure of variation for interval-ratio and ordinal variables; it is equal to the square root of the variance The advantage of the standard deviation is that unlike the variance, it is measured in the same units as the original data. In a distribution where all the scores are identical, the standard deviation is zero (0). Zero is the lowest possible value for the standard deviation; in an identical distribution, all the points would be the same, with the same mean, mode, and median. There is no variation or dispersion in the scores.The more the standard deviation departs from zero, the more variation there is in the distribution. There is no upper limit to the value of the standard deviation.  Range = maximum value - minimum value  IQR = median value of upper half - median value of lower half Standard deviation is the square root of variance  S - for sample SD  Sigma - for population SD Formula: SD = Where x represents an observed value, is the sample mean, and n is the number of observations in the data set.

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Wk 4 Discussion 1 - Measures of Variability due Thurs

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