Identical Values Compute the sample standard deviation of the following test scores: 78, 78, 78, 78. What can be said about a data set in which all the values are identical?

Verified step by step guidance

1. Identify the data set: the test scores are 78, 78, 78, and 78.
2. Calculate the sample mean \( \bar{x} \) using the formula: \[ \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \] where \( n \) is the number of data points and \( x_i \) are the individual values.
3. Compute the squared deviations from the mean for each data point: \[ (x_i - \bar{x})^2 \] Since all values are identical, each deviation will be zero.
4. Calculate the sample variance \( s^2 \) using the formula: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 \] Note that \( n-1 \) is used because this is a sample variance.
5. Find the sample standard deviation \( s \) by taking the square root of the sample variance: \[ s = \sqrt{s^2} \] Since all values are identical, the variance and standard deviation will be zero, indicating no variability in the data.

Key Concepts

Sample Standard Deviation

Sample standard deviation measures the average amount by which data points in a sample differ from the sample mean. It quantifies the spread or variability within the data set, calculated by taking the square root of the average squared deviations from the mean.

Effect of Identical Values on Variability

When all values in a data set are identical, there is no variation among the data points. This means each value equals the mean, resulting in zero deviations and thus a standard deviation of zero, indicating no spread in the data.

Interpretation of Zero Standard Deviation

A standard deviation of zero implies perfect uniformity in the data set, meaning every observation is exactly the same. This indicates no diversity or dispersion, which can be important when assessing consistency or reliability in measurements.