A Sample Of 20 Observations Has A Standard Deviation Of 4

A sample of 20 observations has a standard deviation of 4. The sum of the squared deviations from the sample mean is:

The core question discusses the relationship between the standard deviation of a sample, its size, and the sum of squared deviations from the mean. Specifically, given a sample size of 20 observations with a standard deviation of 4, the task is to determine the sum of squared deviations from the mean.

In statistics, the standard deviation (s) of a sample relates to the sum of squared deviations (SSD) around the mean by the formula:

SSD = (n - 1) * s2

where n is the sample size. The sample standard deviation is calculated as:

s = √(SSD / (n - 1))

Rearranged, the sum of squared deviations is:

SSD = (n - 1) * s2

Given n=20 and s=4, we compute:

SSD = (20-1) (4)2 = 19 16 = 304

Therefore, the sum of squared deviations from the sample mean is 304.

Paper For Above instruction

The problem of understanding the sum of squared deviations in relation to standard deviation is fundamental in descriptive statistics. When analyzing a set of sample data, the measure of dispersion helps quantify the variability among data points. The standard deviation, being the square root of the average squared deviation from the mean, directly relates to the sum of squared deviations (SSD).

To clarify, in a sample of size n, the sum of squared deviations is obtained by multiplying the variance (which is the square of the standard deviation) by (n - 1). This adjustment of (n - 1) instead of n accounts for the degrees of freedom in the calculation of sample variance, ensuring an unbiased estimator of the population variance.

In our specific example, a sample with n=20 observations and a standard deviation of 4, the SSD can be calculated accordingly. Utilizing the formula SSD = (n - 1) s2, we find SSD = 19 16 = 304. The resulting figure, 304, embodies the total squared distance sum of the data points from the mean, reflecting the overall variability within the sample.

This calculation is instrumental for statisticians assessing data dispersion, understanding sampling variability, and performing further inferential statistical analyses such as constructing confidence intervals or hypothesis testing. The relation between variance, standard deviation, and SSD underscores the importance of these measures in summarizing and interpreting data effectively.

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