Quick Math Standard Deviation
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Standard Deviation in Statistics
Introduction
Standard deviation is a measure of the amount of variation or dispersion in a set of values. It indicates how much individual data points differ from the mean (average) of the data set. A low standard deviation means the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.
Formula
For a data set with \( n \) values \( x_1, x_2, x_3, \ldots, x_n \):
1. Calculate the mean (\( \mu \)):
\[
\mu = \frac{1}{n} \sum_{i=1}^{n} x_i
\]
2. Calculate the variance (\( \sigma^2 \)):
\[
\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i – \mu)^2
\]
3. Calculate the standard deviation (\( \sigma \)):
\[
\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i – \mu)^2}
\]
For a sample, the formulas adjust slightly, using \( n-1 \) instead of \( n \) in the variance calculation to account for the sample size:
\[
\sigma^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2
\]
\[
\sigma = \sqrt{\sigma^2} = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i – \bar{x})^2}
\]
Example
Consider the following data set representing the test scores of 5 students: 85, 90, 92, 88, and 95.
1. Calculate the mean (\( \mu \)):
\[
\mu = \frac{85 + 90 + 92 + 88 + 95}{5} = \frac{450}{5} = 90
\]
2. Calculate each score’s deviation from the mean, square it, and find the average:
\[
(85 – 90)^2 = 25
\]
\[
(90 – 90)^2 = 0
\]
\[
(92 – 90)^2 = 4
\]
\[
(88 – 90)^2 = 4
\]
\[
(95 – 90)^2 = 25
\]
\[
\text{Variance } (\sigma^2) = \frac{25 + 0 + 4 + 4 + 25}{5} = \frac{58}{5} = 11.6
\]
3. Calculate the standard deviation (\( \sigma \)):
\[
\sigma = \sqrt{11.6} \approx 3.41
\]
Interpretation
The standard deviation of approximately 3.41 indicates that, on average, the test scores deviate from the mean score (90) by about 3.41 points. This relatively low standard deviation suggests that the scores are fairly close to the mean.
Standard deviation is a crucial statistical tool for understanding the variability within a data set. It helps to interpret the data’s spread and can be used in various fields to make informed decisions based on data analysis.
