Descriptive Statistics: Skewness and the Mean, Median, and Mode
1.3
2008/06/26 11:35:21 GMT-5
2008/07/14 11:17:33.154 GMT-5
Barbara
Illowsky
cnx@cnx.org
Susan
Dean
cnx@cnx.org
Connexions
cnx@cnx.org
elementary
statistics
Consider the following data set:
- 4
- 5
- 6
- 6
- 6
- 7
- 7
- 7
- 7
- 7
- 7
- 8
- 8
- 8
- 9
- 10
This data
produces the histogram shown below. Each interval has width one and each value is located in the middle of an interval.
The histogram displays a symmetrical distribution of data. The mean, the median, and the mode are each 7 for these data. In a perfectly symmetrical distribution, the mean, the median, and the mode are the same.
The histogram for the data:
- 4
- 5
- 6
- 6
- 6
- 7
- 7
- 7
- 7
- 7
- 7
- 8
is skewed to the left.
The mean is 6.3, the median is 6.5, and the mode is 7. Notice that the mean is less than the median and they are both less than the mode. The mean and the median both reflect the skewing but the mean more so.
The histogram for the data:
- 6
- 7
- 7
- 7
- 7
- 7
- 7
- 8
- 8
- 8
- 9
- 10
is skewed to the right.
The mean is 7.7, the median is 7.5, and the mode is 7. Notice that the mean is the largest statistic, while the mode is the smallest. Again, the mean reflects the skewing the most.To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is less than the mode. If the distribution of data is skewed to the right, the mode is less than the median, which is less than the mean. Skewness and symmetry become important when we discuss probability distributions in later chapters.