The grand mean or pooled mean is the

average In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...

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: Base meaning * Sample (statistics), a subset of a population – complete data set * Sample (signal), a digital discrete sample of a continuous analog signal * Sample (material), a specimen or small quantity of so ...

for a

measure Measure may refer to: * Measurement, the assignment of a number to a characteristic of an object or event Law * Ballot measure, proposed legislation in the United States * Church of England Measure, legislation of the Church of England * Mea ...

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. Everitt, B. S. (2006). The Cambridge Dictionary of Statistics (3 ed.). Cambridge University Press.

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 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 collection of statistical models and their associated estimation procedures (such as the "variation" among and between groups) used to analyze the differences among means. ANOVA was developed by the statistician ...

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

, 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