Find the mean, the variance, and the standard deviation of...

Find the mean, the variance, and the standard deviation of the data sets $\{f(x) \mid x=i / 10, i=-10, \ldots, 10\}$ associated with the following functions $f(x)$ :
(a) $3 x+1$,
(b) $x^2$,
(c) $x^3-2 x$,
(d) $e^{-x}$,
(e) $\tan ^{-1} x$.

Key Concepts

Mean

The mean is a measure of central tendency calculated by summing all the values in a data set and then dividing by the number of observations. It provides a single value that represents the 'center' or average of the data, making it essential in summarizing and comparing data sets.

Variance

Variance is a measure of dispersion that quantifies the spread of a set of data points around the mean. It is computed by taking the average of the squared differences between each data value and the mean. This concept is crucial for understanding how much the data varies from the central value.

Standard Deviation

Standard deviation is the square root of the variance and gives a measure of dispersion in the same units as the data. It indicates the typical amount by which the data values differ from the mean, helping to assess the consistency or variability within a data set.

Discrete Data Sets

Discrete data sets consist of finite or countable numbers of values, often obtained from evaluating a function at specific input points. Analyzing such data using descriptive statistics like mean, variance, and standard deviation helps in summarizing the underlying trends and variations.

Function Evaluation in Statistical Analysis

Function evaluation in statistical analysis refers to applying a function to a set of values to generate a new set of data. This process can highlight different statistical characteristics and relationships in the data, making it a valuable tool in both theoretical and applied settings.

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