As a domain expert in statistics, I can provide an in-depth explanation of the conditions under which the standard deviation is zero. The standard deviation is a measure of the amount of variation or dispersion in a set of values. A standard deviation of zero indicates that there is no variability in the data set; in other words, all the values are the same.

Let's start by understanding the formula for calculating the standard deviation. For a population, the standard deviation (σ) is given by the square root of the variance (σ²), which is the average of the squared differences from the mean:

\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
\]

Where \( x_i \) represents each value in the data set, \( \mu \) is the mean of the data set, and \( N \) is the number of values.

For a sample, the formula is slightly different, as it uses \( N - 1 \) in the denominator to correct for bias:

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N - 1}}
\]

Here, \( s \) is the sample standard deviation, \( \bar{x} \) is the sample mean, and \( N \) is the sample size.

Now, let's consider the condition where the standard deviation is zero. If the standard deviation is zero, it implies that the variance (the square of the standard deviation) is also zero. Variance is calculated by summing the squared differences between each data point and the mean, then dividing by the number of observations (for a population) or \( N - 1 \) (for a sample). If this sum is zero, it means that the squared differences are zero for every data point.

Mathematically, this can be expressed as:

\[
\sum (x_i - \mu)^2 = 0
\]

For a sample, it would be:

\[
\sum (x_i - \bar{x})^2 = 0
\]

This can only be true if each term \( (x_i - \mu)^2 \) or \( (x_i - \bar{x})^2 \) is zero. Squaring a real number always results in a non-negative value, and the only real number that, when squared, equals zero is zero itself. Therefore, for the squared difference to be zero, \( x_i \) must be equal to \( \mu \) or \( \bar{x} \) for every \( i \). This means that every data point in the set is identical to the mean.

In conclusion, the standard deviation of a data set is zero if and only if all of its values are identical. This is a fundamental concept in statistics that reflects the complete lack of variation within the data.

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This means that for every i, the term (xi - x )2 = 0. This means that every data value is equal to the mean. This result along with the one above allows us to say that the sample standard deviation of a data set is zero if and only if all of its values are identical.Apr 9, 2018read more >>