The grand mean or pooled mean is the

average In colloquial, ordinary language, an average is a single number or value that best represents a set of data. The type of average taken as most typically representative of a list of numbers is the arithmetic mean the sum of the numbers divided by ...

of the means of several subsamples, as long as the subsamples have the same number of data points. For example, consider several lots, each containing several items. The items from each lot are

sampled Sample or samples may refer to: * Sample (graphics), an intersection of a color channel and a pixel * Sample (material), a specimen or small quantity of something * Sample (signal), a digital discrete sample of a continuous analog signal * Sample ...

for a measure of some variable and the means of the measurements from each lot are computed. The mean of the measures from each lot constitutes the subsample mean. The mean of these subsample means is then the grand mean.

Example

Suppose there are three groups of numbers: group A has 2, 6, 7, 11, 4; group B has 4, 6, 8, 14, 8; group C has 8, 7, 4, 1, 5. The mean of group A = (2+6+7+11+4)/5 = 6, The mean of group B = (4+6+8+14+8)/5 = 8, The mean of group C = (8+7+4+1+5)/5 = 5, Therefore, the grand mean of all numbers = (6+8+5)/3 = 6.333.

Application

Suppose one wishes to determine which states in America have the tallest men. To do so, one measures the height of a suitably sized sample of men in each state. Next, one calculates the means of height for each state, and then the grand mean (the mean of the state means) as well as the corresponding

standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...

of the state means. Now, one has the necessary information for a preliminary determination of which states have abnormally tall or short men by comparing the means of each state to the grand mean ± some multiple of the standard deviation. In

ANOVA Analysis of variance (ANOVA) is a family of statistical methods used to compare the means of two or more groups by analyzing variance. Specifically, ANOVA compares the amount of variation ''between'' the group means to the amount of variation ''w ...

, there is a similar usage of grand mean to calculate sum of squares (SSQ), a measurement of variation. The total variation is defined as the sum of squared differences between each score and the grand mean (designated as GM), given by the equation :

SSQ_ = \sum (X-GM)^2

Discussion

The term ''grand mean'' is used for two different concepts that should not be confused, namely, the overall mean and the mean of means. The overall mean (in a grouped data set) is equal to the

sample mean The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or me ...

, namely,

\frac\sum_^N x_

. The mean of means is literally the mean of the ''G (g=1,...,G)'' group means

\bar_g

, namely,

\frac\sum_^G \bar_g

. If the sample sizes across the ''G'' groups are equal, then the two statistics coincide.

References

{{reflist Descriptive statistics Means

Example

Application

Discussion

See also

References