Variance and standard deviation for ungrouped data calculator

Variance and Standard Deviation of Ungrouped Data Calculator

Use this calculator to find the variance and standard deviation for ungrouped data.

Calculator

How to use Variance and Standard Deviation of Ungrouped Data?

Step 1 - Enter the set of numerical values (X) seperated by ,

Step 2 - Click on Calculate button to calculate sample mean, sample variance and sample standard deviation

Step 3 - Calculate number of observation (n)

Step 4 - Calculate sample mean for ungrouped data

Step 5 - Calculate sample variance for ungrouped data

Step 6 - Calculate sample standard deviation for ungrouped data

Variance and Standard Deviation for Ungrouped Data

Let $x_i, i=1,2, \cdots , n$ be $n$ observations

Variance and Standard Deviation formula for ungrouped data

mean of $X$ is denoted by $\overline{X}$ and is given by $$ \begin{eqnarray*} \overline{X}& =\frac{1}{n}\sum_{i=1}^{n}x_i \end{eqnarray*} $$ Variance of $X$ is denoted by $s_{x}^2$ and is given by $$ \begin{aligned} s_x^2 & =\frac{1}{n-1}\sum_{i=1}^{n}(x_i -\overline{x})^2\\ &=\frac{1}{n-1}\bigg(n\sum_{i=1}^{n} x_i^2-\big(\sum_{i=1}^nx_i\big)^2\bigg) \end{aligned} $$

The standard deviation of $X$ is defined as the positive square root of variance. The standard deviation of $X$ is given by

$$ \begin{eqnarray*} s_x & =\sqrt{s_x^2} \end{eqnarray*} $$

Below are the numerical examples with step by step guide solution on variance and standard deviation for ungrouped data.

Variance and Standard deviation of ungrouped data Example 1

The age (in years) of 6 randomly selected students from a class are

22,25,24,23,24,20.

Find the variance and standard deviation.

Solution

	$x_i$	$x_i^2$
	22	484
	25	625
	24	576
	23	529
	24	576
	20	400
Total	138	3190

Sample mean for ungrouped data

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{138}{6}\\ &=23\text{ years} \end{aligned} $$

The average of age of students is $23$ years.

Sample variance for ungroped data

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{5}\bigg(3190-\frac{(138)^2}{6}\bigg)\\ &=\dfrac{1}{5}\big(3190-\frac{19044}{6}\big)\\ &=\dfrac{1}{5}\big(3190-3174\big)\\ &= \frac{16}{5}\\ &=3.2 \end{aligned} $$

Sample standard deviation of ungrouped data

The standard deviation is the positive square root of the variance.

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{3.2}\\ &=1.7889 \text{ years} \end{aligned} $$

Thus the standard deviation of age of students is $1.7889$ years.

Variance and Standard deviation for ungrouped data Example 2

The following data gives the hourly wage rates (in dollars) of 10 employees of a company.

20,21,24,25,18,22,24,22,20,22.

Find the variance and standard deviation of hourly wage rates.

Solution

	$x_i$	$x_i^2$
	20	400
	21	441
	24	576
	25	625
	18	324
	22	484
	24	576
	22	484
	20	400
	22	484
Total	218	4794

Sample mean of ungrouped data

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{218}{10}\\ &=21.8\text{ dollars} \end{aligned} $$

The average of hourly wage rates is $21.8$ dollars.

Sample variance of ungrouped data

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{9}\bigg(4794-\frac{(218)^2}{10}\bigg)\\ &=\dfrac{1}{9}\big(4794-\frac{47524}{10}\big)\\ &=\dfrac{1}{9}\big(4794-4752.4\big)\\ &= \frac{41.6}{9}\\ &=4.6222 \end{aligned} $$

Sample standard deviation of ungrouped data

The standard deviation is the positive square root of the variance.

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{4.6222}\\ &=2.1499 \text{ dollars} \end{aligned} $$

Thus the standard deviation of hourly wage rates is $2.1499$ dollars.

Variance and Standard deviation of ungrouped data Example 3

A random sample of 15 patients yielded the following data on the length of stay (in days) in the hospital.

5, 6, 9, 10, 15, 10, 14, 12, 10, 13, 13, 9, 8, 10, 12.

Compute variance and standard deviation for given data

Solution

	$x_i$	$x_i^2$
	5	25
	6	36
	9	81
	10	100
	15	225
	10	100
	14	196
	12	144
	10	100
	13	169
	13	169
	9	81
	8	64
	10	100
	12	144
Total	156	1734

Sample mean for ungrouped data

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{156}{15}\\ &=10.4\text{ days} \end{aligned} $$

The average of length of stay in the hospital is $10.4$ days.

Sample variance for ungrouped data

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{14}\bigg(1734-\frac{(156)^2}{15}\bigg)\\ &=\dfrac{1}{14}\big(1734-\frac{24336}{15}\big)\\ &=\dfrac{1}{14}\big(1734-1622.4\big)\\ &= \frac{111.6}{14}\\ &=7.9714 \end{aligned} $$

Sample standard deviation for ungrouped data

The standard deviation is the positive square root of the variance.

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{7.9714}\\ &=2.8234 \text{ days} \end{aligned} $$

Thus the standard deviation of length of stay in the hospital is $2.8234$ days.

Variance and Standard deviation for ungrouped data Example 4

Blood sugar level (in mg/dl) of a sample of 20 patients admitted to the hospitals are as follows:

75,89,72,78,87, 85, 73, 75, 97, 87, 84, 76,73,79,99,86,83,76,78,73.

Calculate variance and standard deviation for the given data.

Solution

	$x_i$	$x_i^2$
	75	5625
	80	6400
	72	5184
	78	6084
	82	6724
	85	7225
	73	5329
	75	5625
	97	9409
	87	7569
	84	7056
	76	5776
	73	5329
	79	6241
	99	9801
	86	7396
	83	6889
	76	5776
	78	6084
	73	5329
Total	1611	130851

Sample mean for ungrouped data

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{1611}{20}\\ &=80.55\text{ mg/dl} \end{aligned} $$

The average of blood sugar level is $80.55$ mg/dl.

Sample variance for ungrouped data

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{19}\bigg(130851-\frac{(1611)^2}{20}\bigg)\\ &=\dfrac{1}{19}\big(130851-\frac{2595321}{20}\big)\\ &=\dfrac{1}{19}\big(130851-129766.05\big)\\ &= \frac{1084.95}{19}\\ &=57.1026 \end{aligned} $$

Sample standard deviation for ungrouped data

The standard deviation is the positive square root of the variance.

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{57.1026}\\ &=7.5566 \text{ mg/dl} \end{aligned} $$

Thus the standard deviation of blood sugar level is $7.5566$ mg/dl.

Variance and Standard deviation of ungrouped data Example 5

The rice production (in Kg) of 10 acres is given as: 1120, 1240, 1320, 1040, 1080, 1720, 1600, 1470, 1750, and 1885. Find variance and standard deviation for the given data.

Solution

	$x_i$	$x_i^2$
	1120	1254400
	1240	1537600
	1320	1742400
	1040	1081600
	1080	1166400
	1720	2958400
	1600	2560000
	1470	2160900
	1750	3062500
	1885	3553225
Total	14225	21077425

Sample mean for ungrouped data

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n x_i\\ &=\frac{14225}{10}\\ &=1422.5\text{ Kg} \end{aligned} $$

The average of rice production is $1422.5$ Kg.

Sample variance for ungrouped data

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\frac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{9}\bigg(21077425-\frac{(14225)^2}{10}\bigg)\\ &=\dfrac{1}{9}\big(21077425-\frac{202350625}{10}\big)\\ &=\dfrac{1}{9}\big(21077425-20235062.5\big)\\ &= \frac{842362.5}{9}\\ &=93595.8333 \end{aligned} $$

Sample standard deviation for ungrouped data

The standard deviation is the positive square root of the variance.

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{93595.8333}\\ &=305.9344 \text{ Kg} \end{aligned} $$

Thus the standard deviation of rice production is $305.9344$ Kg.

Conclusion

Hope you like above article about formula for variance and standard deviation for ungrouped data and how to calculate variance and standard deviation for ungrouped data.

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