One Standard Deviation above the Mean

Mean is the number obtained by dividing the sum of observations of data by the number of observations. Subtracting the mean from the value of an observation gives the "deviation from the mean". Computing mean is of great importance in all branches of practical work
Introduction to standard deviation:

Standard deviation is the square root of the arithmetic mean of the squares of the deviations. It is denoted generally by the Greek letter sigma, `Sigma`   . Formula for finding standard deviation is,

`Sigma = sqrt(((a_1-a_2)^2 + (a_2-a_3)^2 +......+(a_n - a)^2)/n) `

where, `a = (a_1+a_2+.....+a_n)/(n)` , and a1, a2, ....., an are the given numbers,

Standard Deviation formulas for learning is:

S =` sqrt(((sum(x - barx)^2)) / (n-1))`

n = total number of observations, a1 - a2 = a2 - a3 =....= an - a = variance

In Standard deviation, we find the mean for the given data’s, variance from the mean and finally we find the standard deviation from the variance by squaring the variance.
Examples on Standard Deviation

Ex 1:  Calculate the mean and standard deviation for the given data set.
i    x
1    300
2    800
3    900
4    1000
5    500
6    100

Sol:

Step I: `barx` = `( 300 + 800 + 900 + 1000 + 500 + 100) / 6`
                     = 600

Step II:
 `x`        `x - barx`        ` ( x - barx)^2`
300      300  -  600 = -300       90000
800      800  -  600= 200    40000
900      900  - 600 = -300    90000
1000       1000  - 600 = -400    160000
500      500  - 600 = -100       10000
100        100  - 600 = -500    250000



Step III: ` (x - barx)^2` = 640000

Step IV:    S =  `sqrt( 640000 / 5 )`

Step V:     s =  `sqrt( 128000 )`

Step V:  ` S = 357.770`

Ans:    `barx` ` = 600`

`S = 357.770`

Ex 2:  Find the Standard deviation of the given Data 7, 8, 9, 10 and 11

Sol: Step I:  `barx` =` ( 7 + 8 + 9 + 10 + 11 ) / 5`

`barx` = ` 45 / 5`

`barx` = 9

Step III:
x`     `x - barx `(x - barx)^2`
7    7 - 9 = -2    4
8    8 - 9 = -1    1
9    9 - 9 = 0    0
10    10 - 9 = 1    1
11    11 - 9= 2    4


Step IV;  Sum of the `(x - barx)^2`

   ` 4+1+0+1+4 = 10`

Step V: Standard Deviation formulas for learning is:

S =` sqrt(((sum(x - barx)^2)) / (n-1))`

Step V:                    S =` sqrt( ( 4 + 1 + 0 + 1 + 4 ) / 4 )`

S = `sqrt(10 / 4)`

Step VI:    S = `sqrt(2.5)`

                S = 1.581

Ans:      S = 1.581

 

 

Example on Standard Deviation

Ex 3: Find the Standard deviation of the given Data by hand 18, 28 43, 54, 57 and 70.

Sol:  Step I;      Mean  `barx` =` (18 + 28 + 43 + 54 + 57 + 70 ) / 6`

`barx` =` 270 / 6`

`barx` = 45

Step II: Variance:
`x`    `x-barx`    `(x-barx)^2`
18    18 - 45 = -27     729
28    28 - 45 = -17     289
43    43 - 45 = -2     4
54    54 - 45 = 9         81
57    57 - 45 = 12     144
70    70 - 45 = 25     625


Step III:    Sum of the `(x - barx)^2`

` 729 + 289 + 4 + 81 + 144 + 625 = 1872`

Step IV:   Standard Deviation formulas to learn is:

S =` sqrt(((sum(x - barx)^2)) / (n-1))`

Step V:   S = `sqrt ((729 + 289 + 4 + 81 + 144 + 625) / (5-1))`

             S = `sqrt(1872 / 4)`

Step VI:    S = `sqrt(468)`

                S = 21.633

Ans: S = 21.633