What happens to the mean and median if we add or multiply each observation in a data set by a constant?
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!!
Shape
The 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, and mode are all the same here
- no skewness is apparent
- the distribution is described as symmetric
Mean = Median = Mode Symmetrical
Left-Skewed or Skewed Left
- mean < median
- long tail on the left
Median Mean Mode Skewed to the left
Right-skewed or Skewed Right
- mean > median
- long tail on the 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.
3.2Measures of Dispersion1Determine the range, standard deviation, and variance of a variable from raw data.MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.Compute the range for the set of data.1) 9, 10, 11, 12, 13A) 4B) 13C) 0.8D) 92) 10, 10, 10, 18, 29, 29, 29A) 19B) 19.3C) 18D) 90.63) 14, 16, 14, 16, 14, 16, 14, 16
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4) 6, 18, 2, 14, 12A) 16B) 18C) 2D) 6
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6) 7, 8, 4, 2, 6, 11, 9, 9, 8A) 9B) 11C) 7.1D) 137) 74, 142, 35, 103, 199
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8) 111, 521, 225, 654, 354, 272
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9) 2.6, 5.4, 1.2, 4.4, 6.4, 3.8A) 5.2B) 6.4C) 1.2D) 4.0
10) 0.2, 0.11, 0.464, 0.382, 0.598, 0.285
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Find the sample standard deviation.11) 2, 3, 4, 5, 6
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12) 6, 6, 6, 9, 12, 12, 12
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13) 7, 17, 16, 10, 9, 8, 15, 13, 11A) 3.6B) 3.4C) 13.2D) 11.7
14) 42, 87, 69, 60, 64, 22, 67, 74, 49
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15) 232, 225, 104, 285, 208, 209, 268, 265, 128
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16) 18, 11, 12, 10, 7, 10, 5, 9, 12, 25
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