The given values represent data for a sample. Find the...

The given values represent data for a sample. Find the variance and the standard deviation based on this sample.
$\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 33 & {3} \\ {34} & {1} \\ {35} & {4} \\ {36} & {6} \\ {37} & {5} \\ {38} & {11} \\ {39} & {6} \\ \hline\end{array}$

Key Concepts

Frequency Distribution

A frequency distribution is a way of summarizing data that shows the number of observations (frequencies) corresponding to each unique value or range of values. This concept is essential when dealing with data sets where the same value occurs multiple times, as it allows for a clearer and more concise representation of the data, enabling further statistical calculations.

Arithmetic Mean

The arithmetic mean, often known simply as the mean, is the average of a set of numbers. It is calculated by summing all the values and dividing by the total number of observations. In the context of a frequency distribution, the mean is computed by multiplying each value by its frequency, summing these products, and then dividing by the total frequency, which is critical for subsequent variance calculations.

Sample Variance

Sample variance is a measure of the spread or dispersion of a set of sample data. It is calculated by taking the average of the squared differences between each data point and the sample mean, typically using a denominator of one less than the number of observations (n-1) to correct for bias. This measure reveals how far the data points tend to deviate from the mean, providing insight into the variability present within the sample.

Sample Standard Deviation

The sample standard deviation is a statistic that quantifies the amount of dispersion or spread in a sample of values. It is computed as the square root of the sample variance, returning the measure to the same units as the data. As such, the sample standard deviation is widely used to understand the distribution of data around the mean in a way that is more interpretable compared to variance.

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