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. $\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 20 & {21} \\ {25} & {10} \\ {30} & {1} \\ {35} & {2} \\ {40} & {2} \\ {40} & {2} \\ {50} & {4} \\ {50} & {5} \\ {55} & {3} \\ {60} & {7} \\ \hline\end{array}$

The given values represent data for a population. Find the variance and the standard deviation for each set of data.
$\begin{array}{|c|c|}\hline x_{i} & {f_{i}} \\ \hline 20 & {21} \\ {25} & {10} \\ {30} & {1} \\ {35} & {2} \\ {40} & {2} \\ {40} & {2} \\ {50} & {4} \\ {50} & {5} \\ {55} & {3} \\ {60} & {7} \\ \hline\end{array}$

Key Concepts

Weighted Mean

The weighted mean is a measure of central tendency that takes into account the frequency or weight of each data point. When dealing with frequency distributions, each value is multiplied by its corresponding frequency, and the resulting products are summed and then divided by the total frequency. This provides a more accurate representation of the dataset when data values are repeated or when some values have more influence than others.

Population Variance

Population variance is a measure of how much the values in an entire population differ from the mean of the population. It is calculated by taking the weighted sum of the squared differences between each data point and the mean, and then dividing by the total number of observations in the population. In problems involving frequency data, each squared difference is multiplied by the corresponding frequency before summing.

Frequency Distribution

A frequency distribution is a summary of data that shows the number of times each unique data point or range of data points occurs in a dataset. Frequency distributions are useful in statistical analysis as they allow for the computation of weighted measures like the weighted mean, variance, and standard deviation, especially when the data is given in a tabulated form with repeated values.

Population Standard Deviation

The population standard deviation is the square root of the population variance. It provides a measure of the average distance of the data points from the mean, expressed in the same units as the data. This measure is useful for understanding the dispersion of the population and is particularly important when comparing variability across different datasets.

Please give Ace some feedback