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Answer :
The mean, variance, and standard deviation are statistical measures used to describe the central tendency and spread of a data set. The mean is the average of the data, variance is the average of the squared differences from the mean, and standard deviation is the square root of the variance, indicating distribution spread.
Calculating the Mean, Variance, and Standard Deviation
The mean is calculated by summing all the data values and dividing by the number of data points. The variance measures the spread of the data by averaging the squared deviations of each data value from the mean. Standard deviation is the square root of the variance, providing a measure of spread that is in the same units as the data.
Example Calculation
For example, to calculate the standard deviation of a given set of scores, you first order the scores, then calculate the mean score. After determining the deviations from the mean, you square these deviations. The mean of the squared deviations is the variance. Finally, by taking the square root of the variance, you have calculated the standard deviation.
To illustrate how different standard deviations can affect the shape of a distribution, if two normal distributions have the same mean but different standard deviations, the one with the larger standard deviation will be more spread out, while the one with the smaller standard deviation will be more peaked or narrow.
Computing the Sample Mean and Standard Deviation
When computing the sample mean and standard deviation for a set of scores, follow the same process as described. On adding a new value to a dataset (such as 65 to the previous list), the range, variance, and standard deviation can all be affected, generally leading to higher values due to the increased data spread.
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