Standard deviation

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Definition

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.

Examples

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:

[math]\overline{x}=191[/math]

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:
  
  

[math]x_1=152\quad\longrightarrow\quad(152-191)^2=1\,521[/math]

[math]x_2=162\quad\longrightarrow\quad(162-191)^2=841[/math]

[math]x_3=173\quad\longrightarrow\quad(173-191)^2=324[/math]

[math]x_4=188\quad\longrightarrow\quad(188-191)^2=9[/math]

[math]x_5=193\quad\longrightarrow\quad(193-191)^2=4[/math]

[math]x_6=198\quad\longrightarrow\quad(198-191)^2=49[/math]

[math]x_7=203\quad\longrightarrow\quad(203-191)^2=144[/math]

[math]x_8=269\quad\longrightarrow\quad(269-191)^2=6\,084[/math]

• 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]

Attribution

Web Resources

Wikipedia - standard deviation provides a technical definition Wiktionary - standard deviation Statistics Online Computational Resource - Measures of Variation