The grand mean or pooled mean is the
average 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 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
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
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, there is a similar usage of grand mean to calculate
sum of squares In mathematics, statistics and elsewhere, sums of squares occur in a number of contexts:
Statistics
* For partitioning of variance, see Partition of sums of squares
* For the "sum of squared deviations", see Least squares
* For the "sum of squar ...
(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
:
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, namely,
. The mean of means is literally the mean of the ''G (g=1,...,G)'' group means
, namely,
. If the sample sizes across the ''G'' groups are equal, then the two statistics coincide.
See also
*
Pooled variance
References
{{reflist
Descriptive statistics
Means