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Can you explain variance, standard and deviation to me?

Feb 20, 2024

Can you explain variance, standard and deviation to me?

Solution by Steps

step 1

To understand variance, we start with its definition. Variance measures how much the values in a dataset are spread out from their mean. The formula for the variance

\sigma^2

of a population is:

\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}

where

x_i

represents each value in the dataset,

\mu

is the population mean, and

N

is the number of values in the population

step 2

For a sample variance

s^2

, the formula is slightly different to correct for bias in the estimation of a population variance:

s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}

where

x_i

represents each value in the sample,

\bar{x}

is the sample mean, and

n

is the number of values in the sample

step 3

Standard deviation is the square root of variance and provides a measure of spread in the same units as the data. The population standard deviation

\sigma

is:

\sigma = \sqrt{\sigma^2}

and the sample standard deviation

s

is:

s = \sqrt{s^2}

Answer

Variance is the average of the squared differences from the Mean. Standard deviation is the square root of the variance.

Key Concept

Variance and standard deviation measure spread in a dataset.

Explanation

Variance quantifies the degree to which each number in the set is different from the mean. Standard deviation provides a measure of spread in the same units as the data, making it more interpretable.