In a study of reaction times to a specific stimulus, a...

In a study of reaction times to a specific stimulus, a psychologist recorded these data (in seconds). Find the variance and standard deviation for the data. $\begin{array}{lc}{\text { Class limits }} & {\text { Frequency }} \\ \hline 2.1-2.7 & {12} \\ {2.8-3.4} & {13} \\ {3.5-4.1} & {7} \\ {4.2-4.8} & {5} \\ {4.9-5.5} & {2} \\ {5.6-6.2} & {1}\end{array}$

In a study of reaction times to a specific stimulus, a psychologist recorded these data (in seconds). Find the variance and standard deviation for the data.
$\begin{array}{lc}{\text { Class limits }} & {\text { Frequency }} \\ \hline 2.1-2.7 & {12} \\ {2.8-3.4} & {13} \\ {3.5-4.1} & {7} \\ {4.2-4.8} & {5} \\ {4.9-5.5} & {2} \\ {5.6-6.2} & {1}\end{array}$

Key Concepts

Grouped Data

Grouped data refers to data that have been organized into groups or intervals, usually when there are many observations. This method simplifies the data analysis process by summarizing raw data into frequency distributions, making it easier to calculate statistical measures such as the mean, variance, and standard deviation.

Class Midpoint

The class midpoint is the value that lies at the center of a class interval. In the context of grouped data, it is used as a representative value of all data points within that interval. Calculating midpoints involves averaging the lower and upper boundaries of each interval, and these midpoints are then used to approximate the distribution of the data when computing statistics.

Variance

Variance is a statistical measure that quantifies the spread of data points in a dataset. For grouped data, variance is calculated by finding the weighted average of the squared differences between each class midpoint and the overall mean, with weights corresponding to the frequencies of the respective intervals. It provides a measure of the dispersion of the dataset around the mean.

Standard Deviation

Standard deviation is a measure of dispersion that indicates how much the data points in a dataset deviate from the mean on average. It is the square root of the variance and is expressed in the same units as the original data, making it easier to interpret. For grouped data, after computing the variance using class midpoints and frequencies, the standard deviation is obtained by taking the square root of the variance.

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