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

The given values represent data for a population. Find the variance and the standard deviation for each set of data. $\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 30 & {1} \\ {35} & {7} \\ {40} & {10} \\ {45} & {9} \\ {50} & {9} \\ {55} & {8} \\ {60} & {6} \\ \hline\end{array}$

The given values represent data for a population. Find the variance and the standard deviation for each set of data.
$\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 30 & {1} \\ {35} & {7} \\ {40} & {10} \\ {45} & {9} \\ {50} & {9} \\ {55} & {8} \\ {60} & {6} \\ \hline\end{array}$

Key Concepts

Population Data

Population data includes every member of a complete set of observations under study, meaning all data points are accounted for, which allows for the calculation of true, exact parameters rather than estimates. This distinguishes it from a sample that represents only a part of the entire set.

Frequency Distribution

A frequency distribution organizes data values alongside their frequencies (or counts of occurrence) in the dataset. It is essential in weighted calculations, such as calculating the mean or variance for data that appears with different frequencies.

Weighted Mean

The weighted mean is the average value of a dataset where each data point contributes proportionally to its frequency or weight. It is particularly important when data values are repeated or have varying degrees of significance, as in a frequency distribution.

Variance

Variance quantifies the dispersion of data points by measuring the average squared deviation from the mean. In population data, variance is computed by taking the weighted average of the squared differences between each data value and the mean, providing insight into the data’s spread.

Standard Deviation

Standard deviation is the square root of the variance and offers a measure of dispersion that is in the same units as the data itself. It indicates how much the individual data points deviate from the mean on average, making it a key statistical metric for understanding data variability.

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