Empirical Rule Calculator with examples

Empirical Rule Calculator for grouped data

Empirical rule is the general rule of thumb that applies to bell shaped (symmetrical) distribution. The empirical rule can be stated as :

• $68$% of the data will fall within one standard deviation of the mean,
• $95$% of the data will fall within two standard deviations of the mean,
• $99.7$% of the data will fall within three standard deviations of the mean.

Let $(x_i,f_i), i=1,2, \cdots , n$ be given frequency distribution. If the distribution of $x$ is approximately symmetrical, then

$68$% of the data falls in $\overline{x}\pm 1 s_x$

$95$% of the data falls in $\overline{x}\pm 2 s_x$

$99.7$% of the data falls in $\overline{x}\pm 3 s_x$

where,

• $\overline{x}=\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i$ is the sample mean,
• $s_x =\sqrt{\dfrac{1}{N-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\dfrac{\big(\sum_{i=1}^n f_ix_i\big)^2}{N}\bigg)}$ is the sample standard deviation.

Using the Empirical Rule calculator for grouped data provides step by step guide procedure to calculate empirical rule for frequency distribution.

Empirical Rule Calculator

How to use or setup Empirical Rule Calculator for grouped data

Step 1 - Select type of frequency distribution either Discrete or continuous

Step 2 - Enter the Range or classes (X) seperated by comma (,)

Step 3 - Enter the Frequencies (f) seperated by comma

Step 4 - Click on “Calculate” for Empirical Rule Calculator for grouped data

Step 6 - Gives output as number of observation (n)

Step 7 - Calculate empirical rule mean

Step 8 - Calculate empirical rule standard deviation

Step 9 - Calculate 68% of the data falls within one standard deviation of the mean

Step 10 - Calculate 95% of the data falls within two standard deviations of the mean

Step 11 - Calculate 99.7% of the data falls within three standard deviations of the mean

Lets check out few examples with step by step explaination on using Empirical Rule Calculator for grouped data.

Example - 1 Using Empirical Rule Calculator for frequency distribution

A librarian keeps the records about the amount of time spent (in minutes) in a library by college students. Data is as follows:

Time spent	30	32	35	38	40
No. of students	8	12	20	10	5

Calculate Empirical rule for the above frequency distribution.

Solution

	$x_i$	$f_i$	$f_i*x_i$	$f_ix_i^2$
	30	8	240	7200
	32	12	384	12288
	35	20	700	24500
	38	10	380	14440
	40	5	200	8000
Total		55	1904	66428

Empirical Rule formula Calculator with sample mean

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{n}\sum_{i=1}^n f_ix_i\\ &=\frac{1904}{55}\\ &=34.6182\text{ minutes} \end{aligned} $$

The average time spent in library is $34.6182$ minutes.

Sample variance

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\frac{\big(\sum_{i=1}^n f_ix_i\big)^2}{n}\bigg)\\ &=\dfrac{1}{54}\bigg(66428-\frac{(1904)^2}{55}\bigg)\\ &=\dfrac{1}{54}\big(66428-\frac{3625216}{55}\big)\\ &=\dfrac{1}{54}\big(66428-65913.01818\big)\\ &= \frac{514.98182}{54}\\ &=9.5367 \end{aligned} $$

Empirical Rule Calculator with Sample standard deviation

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

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{17}\\ &=3.0882 \text{ minutes} \end{aligned} $$

Thus the standard deviation of time spent in library is $3.0882$ minutes.

Empirical Rule

$68$% of the students spent time in the library between

$$ \begin{aligned} & \overline{x}- 1 s_x \text{ and } \overline{x}+ 1 s_x \text{ minutes}\\ \Rightarrow & 34.6182 - 1* 3.0882 \text{ and } 34.6182 + 1* 3.0882 \text{ minutes}\\ \Rightarrow & 31.53 \text{ and } 37.7064 \text{ minutes}\\ \end{aligned} $$

$95$% of the students spent time in the library between

$$ \begin{aligned} & \overline{x}- 2 s_x \text{ and } \overline{x}+ 2 s_x \text{ minutes}\\ \Rightarrow & 34.6182 - 2* 3.0882 \text{ and } 34.6182 + 2* 3.0882 \text{ minutes}\\ \Rightarrow & 28.4418 \text{ and } 40.7946 \text{ minutes}\\ \end{aligned} $$

$99.7$% of the students spent time in the library between

$$ \begin{aligned} & \overline{x}- 3 s_x \text{ and } \overline{x}+ 3 s_x \text{ minutes}\\ \Rightarrow & 34.6182 - 3* 3.0882 \text{ and } 34.6182 + 3* 3.0882 \text{ minutes}\\ \Rightarrow & 25.3536 \text{ and } 43.8828 \text{ minutes}\\ \end{aligned} $$

Example - 2 Empirical Rule Calculator

The following table gives the amount of time (in minutes) spent on the internet each evening by a group of 56 students.

Time spent on Internet ($x$)	10-12	13-15	16-18	19-21	22-24
No. of students ($f$)	3	12	15	24	2

Check Empirical rule for the above frequency distribution.

Solution

Let $X$ denote the time spent on the internet.

Here the classes are inclusive. To make them exclusive type subtract 0.5 from the lower limit and add 0.5 to the upper limit of each class.

	Class Interval	Class Boundries	mid-value ($x_i$)	Freq ($f_i$)	$f_i*x_i$	$f_ix_i^2$
	10-12	9.5-12.5	11	3	33	363
	13-15	12.5-15.5	14	12	168	2352
	16-18	15.5-18.5	17	15	255	4335
	19-21	18.5-21.5	20	24	480	9600
	22-24	21.5-24.5	23	2	46	1058
Total				56	982	17708

Empirical Rule Calculator with Sample mean

The sample mean of $X$ is

$$ \begin{aligned} \overline{x} &=\frac{1}{N}\sum_{i=1}^n f_ix_i\\ &=\frac{982}{56}\\ &=17.5357\text{ minutes} \end{aligned} $$

The average time spent on the internet is $17.5357$ minutes.

Sample Variance

Sample variance of $X$ is

$$ \begin{aligned} s_x^2 &=\dfrac{1}{N-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\frac{\big(\sum_{i=1}^n f_ix_i\big)^2}{N}\bigg)\\ &=\dfrac{1}{55}\bigg(17708-\frac{(982)^2}{56}\bigg)\\ &=\dfrac{1}{55}\big(17708-\frac{964324}{56}\big)\\ &=\dfrac{1}{55}\big(17708-17220.07143\big)\\ &= \frac{487.92857}{55}\\ &=8.8714 \end{aligned} $$

Empirical Rule formula Calculator with Sample standard deviation

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

The sample standard deviation is

$$ \begin{aligned} s_x &=\sqrt{s_x^2}\\ &=\sqrt{22.5}\\ &=2.9785 \text{ minutes} \end{aligned} $$

Thus the standard deviation of time spent on the internet is $2.9785$ minutes.

Empirical Rule

$68$% of the students spent time on the internet between

$$ \begin{aligned} & \overline{x}- 1 s_x \text{ and } \overline{x}+ 1 s_x \text{ minutes}\\ \Rightarrow & 17.5357 - 1* 2.9785 \text{ and } 17.5357 + 1* 2.9785 \text{ minutes}\\ \Rightarrow & 14.5572 \text{ and } 20.5142 \text{ minutes}\\ \end{aligned} $$

$95$% of the students spent time on the internet between

$$ \begin{aligned} & \overline{x}- 2 s_x \text{ and } \overline{x}+ 2 s_x \text{ minutes}\\ \Rightarrow & 17.5357 - 2* 2.9785 \text{ and } 17.5357 + 2* 2.9785 \text{ minutes}\\ \Rightarrow & 11.5787 \text{ and } 23.4927 \text{ minutes}\\ \end{aligned} $$

$99.7$% of the students spent time on the internet between

$$ \begin{aligned} & \overline{x}- 3 s_x \text{ and } \overline{x}+ 3 s_x \text{ minutes}\\ \Rightarrow & 17.5357 - 3* 2.9785 \text{ and } 17.5357 + 3* 2.9785 \text{ minutes}\\ \Rightarrow & 8.6002 \text{ and } 26.4712 \text{ minutes}\\ \end{aligned} $$

Conclusion

Hope you like and find above article on using Empirical Rule Calculator for grouped data helpful and educational.

Read more about other Statistics Calculator on below links

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