Mean (Arithmetic Average)

The Mean, commonly referred to as the Arithmetic Average, is one of the most widely used measures of central tendency. It provides a single value that represents the overall level or typical value of a data set. For UGC NET Commerce aspirants, understanding the mean is essential, as it forms the foundation for more advanced statistical tools and concepts.

What is Mean?

Definition:
The mean is the sum of all values in a dataset divided by the number of observations. It gives an overall average and assumes that each data point contributes equally.

Formula (for individual data):

$$\bar{x} = \frac{\sum x}{n}$$

Where:

• $\bar{x}$= Mean

• $\sum x$ = Sum of all data values

• $n$ = Number of observations

Example:
Marks scored by 5 students: 40, 50, 60, 70, 80

$$\bar{x} = \frac{40 + 50 + 60 + 70 + 80}{5} = \frac{300}{5} = 60$$

Types of Mean

a. Simple Arithmetic Mean

This is the regular average, used when all data points are given equal importance.

Formula (again):

$$\bar{x} = \frac{\sum x}{n}$$

b. Weighted Arithmetic Mean

Used when different data values carry different importance or weights.

Formula:

$$\bar{x}_w = \frac{\sum (w \cdot x)}{\sum w}$$

Where:

• $w$ = weight assigned to each value

• $x$ = individual data value

Example:
Suppose a student scored:

• 70 in a test (weight 2)

• 80 in assignment (weight 1)

• 90 in final exam (weight 3)

$$\bar{x}_w = \frac{(70 \times 2) + (80 \times 1) + (90 \times 3)}{2 + 1 + 3} = \frac{140 + 80 + 270}{6} = \frac{490}{6} ≈ 81.67$$

Mean for Different Data Series

a. Individual Series

When raw values are given without frequency.

Formula:

$$\bar{x} = \frac{\sum x}{n}$$

b. Discrete Series

When data values and their frequencies are provided.

Formula:

$$\bar{x} = \frac{\sum (f \cdot x)}{\sum f}$$

Where:

• $f$ = frequency

• $x$ = data value

Example:

$$\bar{x} = \frac{(10 \times 2) + (20 \times 3) + (30 \times 5)}{2+3+5} = \frac{20 + 60 + 150}{10} = 23$$

c. Continuous Series (Grouped Data)

For data grouped in class intervals.

Steps:

1. Find mid-point ($m$) of each class: $m = \frac{\text{Lower limit} + \text{Upper limit}}{2}$

2. Use the formula:

$$\bar{x} = \frac{\sum (f \cdot m)}{\sum f}$$

Short-Cut Method for Mean (Assumed Mean Method)

Useful when data values are large.

Formula:

$$\bar{x} = A + \frac{\sum (f \cdot d)}{\sum f}$$

Where:

• $A$ = Assumed mean

• $d=x-\mathrm{A\; (deviation\; from\; assumed\; mean)}$

Merits of Mean

• Simple and easy to compute.

• Based on all observations.

• Suitable for further statistical analysis.

• Uniquely defined for a dataset.

Demerits of Mean

• Affected by extreme values (outliers).

• Cannot be used for qualitative data (e.g., colors, preferences).

• May not be a value present in the dataset.

• Not suitable for skewed distributions.

Mean vs. Other Measures

Practical Applications of Mean

• Financial analysis: Average income, expenses, sales.

• Education: Average marks or performance.

• Economics: Per capita income, average productivity.

• Business forecasting: Sales and demand estimation.

Conclusion

The Mean is a foundational concept in business statistics and is frequently used in analysis and reporting. While it is simple and informative, its limitations especially its sensitivity to extreme values should be kept in mind.