What happens to the mean and median if we add or multiply each observation in a data set by a constant? Show
Consider for example if an instructor curves an exam by adding five points to each student’s score. What effect does this have on the mean and the median? The result of adding a constant to each value has the intended effect of altering the mean and median by the constant. For example, if in the above example where we have 10 aptitude scores, if 5 was added to each score the mean of this new data set would be 87.1 (the original mean of 82.1 plus 5) and the new median would be 86 (the original median of 81 plus 5). Similarly, if each observed data value was multiplied by a constant, the new mean and median would change by a factor of this constant. Returning to the 10 aptitude scores, if all of the original scores were doubled, the then the new mean and new median would be double the original mean and median. As we will learn shortly, the effect is not the same on the variance! Looking Ahead!Why would you want to know this? One reason, especially for those moving onward to more applied statistics (e.g. Regression, ANOVA), is the transforming data. For many applied statistical methods, a required assumption is that the data is normal, or very near bell-shaped. When the data is not normal, statisticians will transform the data using numerous techniques e.g. logarithmic transformation. We just need to remember the original data was transformed!! ShapeThe shape of the data helps us to determine the most appropriate measure of central tendency. The three most important descriptions of shape are Symmetric, Left-skewed, and Right-skewed. Skewness is a measure of the degree of asymmetry of the distribution. Symmetric
Mean = Median = Mode Symmetrical Left-Skewed or Skewed Left
Median Mean Mode Skewed to the left Right-skewed or Skewed Right
Median Mean Mode Skewed to the right Note! When one has very skewed data, it is better to use the median as measure of central tendency since the median is not much affected by extreme values. If the following scores were population scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What would the standard deviation be? Consider the following scores: 21, 22, 22, 0.1, 20, 25, 28, 26, 23, 19, 0.5. Which of the measures listed below would give the best description of the central tendency of these scores? A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The mode(s) for this distribution is(are) _________. Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What is the mean? Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What is the value of mc050-1.jpg? A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The standard deviation for this distribution is _________. Assume sample scores. Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What is the value of mc049-1.jpg? Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What is the range? A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The mode(s) for this distribution is(are) _________. Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What is the value of mc049-1.jpg? If the following scores were population scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What would the standard deviation be? A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The mean for this distribution is _________. Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 What is the median? A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The mean for this distribution is _________. Given the following set of scores X: 10, 12, 6, 8, 9, 11, 13, 13, 5, 0, 1 A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The range for this distribution is _________. A history professor gives a quiz to his class and records the following scores: 13, 11, 11, 9, 12, 13, 16, 14, 11, 10, 8, 13, 20. The variance for this distribution is _________. Assume sample scores. What measure of central tendency is most sensitive to skewness?Of the three measures of tendency, the mean is most heavily influenced by any outliers or skewness. In a symmetrical distribution, the mean, median, and mode are all equal. In these cases, the mean is often the preferred measure of central tendency.
Which measure of central tendency is sensitive to the values of each and every score?However, in this situation, the mean is widely preferred as the best measure of central tendency because it is the measure that includes all the values in the data set for its calculation, and any change in any of the scores will affect the value of the mean.
Is median a sensitive statistic?To learn more about incomes and their right-skewed distributions, read my post about Global Income Distributions. Statisticians say that the median is a robust statistical while the mean is sensitive to outliers and skewed distributions.
Which of the following measures of central tendency is sensitive to extremely high or low scores in a distribution?The mean is sensitive to all scores in a sample (every number in the data affects the mean), which makes it a more "powerful" measure than the median or mode. The mean's sensitivity to all scores also makes it sensitive to extreme values, which is why the median is used when there are extreme values.
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