Average Value & Calculation

Introduction

Average is a single value that represents the complete group of values.

Example: Average mark scored in a class is 80 %, average height in a country, average life span, average temperature in a particular area, etc.

Average is classified into two groups majorly: They are mathematical average or mean and positional average. To find the positional average we can use median and mode.

Average

An average is the central value of the given set of values. Also, on average, the numerator is the sum of all given values, and the denominator is the total number of given values. The average of finite consecutive numbers is always its middle value. The formula to find the average −

$$\mathrm{Average =\frac{sum\: of\: all\: values}{total\: number\: of\: values}}$$

To find the mathematical average or mean, there are three methods, they are arithmetic mean, geometric mean and harmonic mean.

Arithmetic mean

Among the other methods, Arithmetic mean is the most common method used to find the average. Arithmetic mean is classified into two types, they are Simple arithmetic mean, Weighted arithmetic mean.

Simple arithmetic mean

In statistics, the data are collected, recorded, organised and analysed. Each observation uses a direct or shortcut method in arithmetic mean. The observations are classified into three types, they are individual, discrete and continuous series. The formula to find each type is mentioned below −

Individual series

• Direct method − $\mathrm{\bar{x}=\frac{X_1+ X_2+X_3+...+X_n }{N}=\frac{\sum X}{N}}$

  where X¹+ X²+X³+...+Xⁿ = sum of the values

  N = Total count of the values.

• Shortcut method $\mathrm{\bar{x}=A + \frac{\sum d}{N}}$

  where A = assumed value

  d = (x - A)

Discrete series

• Direct method − $\mathrm{ \bar{x}=\frac{\sum fX}{N}}$

  where f = frequency

• Shortcut method − $\mathrm{ \bar{x}=A + \frac{\sum fd}{N}}$

Continuous series

• Direct method − $\mathrm{\bar{x}=\frac{\sum fm}{N}}$

  where m = midpoint

• Shortcut method − $\mathrm{ \bar{x}=A + \frac{\sum fd}{N}}$

  Where d = (m - A)

Weighted arithmetic mean

In simple arithmetic, the weightage is given equally to all the values, whereas in weighted arithmetic the values have different weightage. The formula is given by

$$\mathrm{\bar{x_w}=\frac{W_1 X_1+ W_2 X_2+.....+W_n X_n }{W_1 +W_2+....+W_n} \: (or)\: \bar{x_w} = \frac{\sum WX}{ΣW}}$$

Where X = variable value

W = weights of each variable

Solved examples

1)In an exam the weightage given for each subject is different. Among 3 students only one who scores the highest will get the scholarship. Now from the given data find the student who will get the scholarship. In each subject the weightage is given as Maths - 5, Physics - 3, Chemistry - 2

Subject	Student A	Student B	Student C
Maths	80	89	79
Physics	96	92	95
Chemistry	89	90	87

Calculate $\mathrm{\underline{X_w}}$ from the given data.

To calculate the weighted arithmetic mean of student A,

Using the formula,

$$\mathrm{\bar{x_{wA}}=\frac{W_1 X_1+ W_2 X_2+.....+W_n X_n}{W_1 +W_2+....+W_n }}$$

$$\mathrm{\bar{x_{wA}}=\frac{(5×80) + (3×96)+(2×89)}{5 + 3+ 2}}$$

$$\mathrm{= \frac{866}{10} = 86.6}$$

To calculate the weighted arithmetic mean of student B,

Using the formula,

$$\mathrm{\bar{x_{wB}}=\frac{W_1 X_1+ W_2 X_2+.....+W_n X_n}{W_1 +W_2+....+W_n }}$$

$$\mathrm{\bar{x_{wB}}=\frac{(5×89) + (3×92)+(2×90) }{5 + 3+ 2}}$$

$$\mathrm{=\frac{901}{10}=90.1}$$

To calculate the weighted arithmetic mean of student C,

Using the formula,

$$\mathrm{\bar{x_{wC}}=\frac{W_1 X_1+ W_2 X_2+.....+W_n X_n}{W_1 +W_2+....+W_n }}$$

$$\mathrm{\bar{x_{wC}}=\frac{(5×79) + (3×95)+(2×87) }{5 + 3+ 2 }}$$

$$\mathrm{=\frac{854}{10}=85.4}$$

From the weighted arithmetic mean student B will get the scholarship.

2)Find the arithmetic mean of the following data using the assumed mean method.

Class interval	0 - 5	5 - 10	10 - 15	15 - 20
frequency	14	12	16	17

Answer:

class interval	fi	midpoint xi	di= xi-A A = 10	Σfi di
0 - 6	14	3	- 7	- 98
6 - 12	12	9	- 1	- 12
12 - 18	16	15	5	80
18 - 24	17	21	11	187
	Σfi= 59			Σfi di= 157

The formula is $\mathrm{\bar{x}=A+ \frac{\sum f_i d_i}{\sum f_i}}$

$\mathrm{\bar{x}=10+ \frac{157}{59}}$

$\mathrm{\bar{x}=10+ 2.66}$

$\mathrm{12.66}$

3)Find the Arithmetic mean of the given data using the direct method.

Class interval	2 - 11	12 - 20	21 - 29	30 - 38
frequency	9	6	8	7

Answer:

The difference between each set of class intervals are 1. To make the intervals continuous subtract the upper limit with 0.5 and add the lower limit with 0.5

C. I	C. I improved	f	m	fm
2 - 11	2.5 - 11.5	9	7	63
12 - 20	11.5 - 20.5	6	16	96
21 - 29	20.5 - 29.5	8	25	200
30 - 38	29.5 - 38.5	7	34	238
		Σf = 30		Σfm =597

The formula is $\mathrm{\bar{x}=\frac{\sum fm}{N}}$

$$\mathrm{\bar{x}=\frac{597}{30}}$$

$\mathrm{\bar{x}=19.9}$

4)The mean of 25 scores in a cricket match are recorded as 52. Later it was found that two scores of 6 were misrecorded as 4 and 5 were misrecorded as 2. Find the correct mean.

Answer:

Given data, N = 25

$$\mathrm{\bar{x}=52}$$

We know, $\mathrm{\bar{x}=\frac{\sum x}{N}}$

$$\mathrm{\sum x = \bar{x}. N}$$

Σx = 25 × 52 = 1300

where Σx is the incorrect mean

Correct Σx = Incorrect Σx - wrong items + correct items

= 1300 - 4 - 2 + 6 + 5

= 1305

Correct $\mathrm{\bar{x}=\frac{1305}{25}= 52.2}$

Conclusion

A data is a group of numbers which contains scientific measurements, surveys and other data collections. An average is a value which represents the collection of values or terms. Arithmetic mean is the method frequently used to find the average when the values are bigger in the arithmetic mean. To make it simple, we use the assumed mean method or shortcut method.

FAQs

1. What is the difference between discrete and continuous function?

In discrete functions, the values are not connected and easily measurable.

Example: Number students want to play football = 23

In continuous functions, the values are connected and have the data in a range.

Example: In a class students scored in the exam in the range 70 - 85

2. When does the simple arithmetic mean and the weighted arithmetic mean give the same result?

When all the given weights are equal then the simple arithmetic mean is equal to the weighted arithmetic mean. we get $\mathrm{\bar{x_w}=\bar{x}}$ where each

$$\mathrm{w_1 = w_2= w_n.}$$

3. What happens when lesser weightage is applied to the higher values?

When less weightage is given to the higher values the value of simple arithmetic mean is greater than the weighted arithmetic mean.

$$\mathrm{i.e.,\bar{x} > \bar{x_w}}$$

4. How to find the average of consecutive series of numbers?

The consecutive numbers are continuous numbers following any series, such as the list of odd numbers upto 20, first 20 natural numbers, etc. To find the average, the centre or middle number is the average.

5. In which places geometric mean and harmonic mean can be applied?

Geometric mean is applied in place of annual growth, and harmonic mean is applied to find the mean of speed.