Standard deviation

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Standard Deviation
  • quantifies the spread of a distribution of data points by measuring how far the points are from their mean, [math]\overline{x}[/math].
  • is the average (or typical distance) between a data point and the mean, [math]\overline{x}[/math].

A low standard deviation indicates that the data points are clustered around the mean value, whereas a high standard deviation indicates that the data points are widely spread with points significantly higher and lower than the mean, [math]\overline{x}[/math].

In most real-life situations, the standard deviation is estimated based on a sample taken from the population. There are many notations for the sample standard deviation: [math]SD, S, Sd, StDev. \,[/math] The sample standard deviation is mathematically defined as:

[math] SD = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}\,, [/math]

where [math]\{x_1,\,x_2,\,\ldots,\,x_n\}[/math] are the data points in the sample and [math]\overline{x}[/math] is the mean of the sample.


Suppose we are interested in the long-jump performance of young adult males. We design an experiment by randomly selecting 100 male students, aged 18-22, to perform the standing long jump. For ease of calculations in this example, we will use the distanced jumped for 8 of the 100 students:

Long-Jump (centimeters) Sample Data
152 162 173 188 193 198 203 269

To find the standard deviation of these 8 distances:

1. Calculate the mean of the 8 data points:


2. Calculate the sum of the squared differences of each data point and the mean, [math]\sum_{i=1}^8 {(x_i-191)^2} [/math]:

  • The squared differences for each data point:


  • The sum of the squared differences:

[math]\sum_{i=1}^8 {(x_i-191)^2}=1\,521+841+324+9+4+49+144+6\,084=8\,976[/math]

3. Divide the resulting sum by [math]n-1[/math] and take the square root of the result:

[math]SD =\sqrt{{1 \over 8-1}\cdot 8\,976} = 35.929[/math]