Handbook of Descriptive Statistics/Measures of Central Tendency/Arithmetic Mean

Description
Most commonly used measure of central tendency.


 * $$\overline{x} = \frac1n\sum_{i=1}^n x_i$$

Wikipedia article: Arithmetic mean

Usages

 * The mean may be conceived of as an estimate of the median. When the mean is not an accurate estimate of the median, the set of numbers, or frequency distribution, is said to be skewed.


 * The arithmetic mean is greatly influenced by outliers. For instance, reporting the "average" annual income in Redmond, Washington as the arithmetic mean of all annual incomes would yield a surprisingly high number because of Bill Gates. These distortions occur when the mean is different from the median, and the median is a superior alternative when that happens.


 * In certain situations, the arithmetic mean is the wrong concept of "average" altogether. For example, if a stock rose 10% in the first year, 30% in the second year and fell 10% in the third year, then it would be incorrect to report its "average" increase per year over this three year period as the arithmetic mean (10% + 30% + (&minus;10%))/3 = 10%; the correct average in this case is the geometric mean which yields an average increase per year of only 8.8%.

Distributions

 * Include standard values for common distributions, if they exist. For example, a normal distribution always has a skew and kurtosis of zero.
 * Include standard error for normal distribution, and for other distributions also if possible.

For a normal distribution, the standard error of the mean is:


 * &sigma;M = &sigma; / sqrt(N)


 * Sampling distribution.

Calculation
Include any alternative methods of calculation, especially for large data sets, or when other measures have already been calculated.

Microsoft Excel
It is relatively easy to determine the arithmetic mean, or average, in Microsoft Excel (v. '97, 2000, 2002/XP) by using a simple equation. Follow these steps to determine the mean in Excel for a set of data:


 * 1) Highlight an empty cell in the worksheet in which the mean will be calculated.
 * 2) Assuming that there are ten data points in cells A1 through A10, the following equation could be typed in the empty cell: "=AVERAGE(A1:A10)". Alternatively, this equation can also accept arguments in the form: "=AVERAGE(A1,A2,A3,...,A10)"

Refer to Microsoft Excels application documentation for further help.

Minitab
In Minitab (v. 14), finding the arithmetic mean is as simple as choosing options from the menu to display several descriptive statistics at once, including the arithmetic mean. Assuming again that there are ten valid data points in column C1, follow the steps below to display the mean:


 * 1) Go to the Menu bar at the top of the Minitab application and choose the following options: Stat -> Basic Statistics -> Display Descriptive Statistics...
 * 2) In the box labeled "Variables:" on the right hand side of the dialog box that appears, type "C1" and then click on the "OK" button in the bottom center of the dialog box. The mean will then be displayed in addition to other useful statistics in the 'Session' window of Minitab.