Standard deviation
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Standard Deviation
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 reallife 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:

Examples
Suppose we are interested in the longjump performance of young adult males. We design an experiment by randomly selecting 100 male students, aged 1822, 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:
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_i191)^2} [/math]:
 The squared differences for each data point:
 [math]x_1=152\quad\longrightarrow\quad(152191)^2=1\,521[/math]
 [math]x_2=162\quad\longrightarrow\quad(162191)^2=841[/math]
 [math]x_3=173\quad\longrightarrow\quad(173191)^2=324[/math]
 [math]x_4=188\quad\longrightarrow\quad(188191)^2=9[/math]
 [math]x_5=193\quad\longrightarrow\quad(193191)^2=4[/math]
 [math]x_6=198\quad\longrightarrow\quad(198191)^2=49[/math]
 [math]x_7=203\quad\longrightarrow\quad(203191)^2=144[/math]
 [math]x_8=269\quad\longrightarrow\quad(269191)^2=6\,084[/math]
 [math]x_1=152\quad\longrightarrow\quad(152191)^2=1\,521[/math]
 The sum of the squared differences:
 [math]\sum_{i=1}^8 {(x_i191)^2}=1\,521+841+324+9+4+49+144+6\,084=8\,976[/math]
 [math]\sum_{i=1}^8 {(x_i191)^2}=1\,521+841+324+9+4+49+144+6\,084=8\,976[/math]
3. Divide the resulting sum by [math]n1[/math] and take the square root of the result:
 [math]SD =\sqrt{{1 \over 81}\cdot 8\,976} = 35.929[/math]
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