Measures of Central Tendency: Mean, Median, Mode and Applications Questions

Question 1

Define arithmetic mean.

Accepted Answer

Arithmetic mean is the sum of all observations divided by the number of observations.

Question 2

Define median.

Accepted Answer

Median is the middle value that divides the ordered data into two equal halves.

Question 3

Define mode.

Accepted Answer

Mode is the value (or class) which occurs with the highest frequency.

Question 4

Write mean formula for individual series.

Accepted Answer

Mean (x̄) = Σx / n.

Question 5

Write mean formula for discrete frequency series.

Accepted Answer

Mean (x̄) = Σfx / Σf.

Question 6

State one merit of median.

Accepted Answer

Median is not affected by extreme values (any one merit).

Question 7

Differentiate mean, median and mode (any three points).

Accepted Answer

Mean, median and mode differ as follows:

| Point | Mean | Median | Mode |
|---|---|---|---|
| Basis | arithmetic average | middle value | most frequent value |
| Uses | further calculations | skewed/outlier data | typical/most common |
| Affected by extremes | yes | no | usually no |

(Any three points can be written.)

Question 8

Write any three merits of arithmetic mean.

Accepted Answer

Merits of arithmetic mean:

1) Easy to understand and calculate.
2) Uses all observations.
3) Suitable for further algebraic/statistical analysis.

(Any three merits can be written.)

Question 9

Write any three demerits of arithmetic mean.

Accepted Answer

Demerits of arithmetic mean:

1) Highly affected by extreme values (outliers).
2) Not suitable for qualitative data.
3) May not represent typical value in skewed distributions.

(Any three demerits can be written.)

Question 10

Explain weighted mean with formula (any three points).

Accepted Answer

Weighted mean is used when items have different importance.

1) Assign weights (w) to each value (x).
2) Compute Σwx and Σw.
3) Weighted mean x̄w = Σwx / Σw.

(Any three points can be written.)

Question 11

Explain median for individual series (any three points).

Accepted Answer

To find median for individual series:

1) Arrange data in ascending order.
2) If n is odd, median is (n+1)/2th item.
3) If n is even, median is average of n/2th and (n/2+1)th items.

(Any three points can be written.)

Question 12

Write the empirical relationship among mean, median and mode.

Accepted Answer

Empirical relationship (for moderately skewed data):

Mode ≈ 3 Median − 2 Mean.

Question 13

Explain arithmetic mean and weighted mean with formulas and a worked example.

Accepted Answer

Arithmetic mean is the most commonly used average. It is obtained by dividing the sum of all observations by the number of observations.

Formulas:

- Individual series: x̄ = Σx / n
- Discrete frequency series: x̄ = Σfx / Σf

Weighted mean is used when all items do not have equal importance. In that case, weights are assigned and average is computed using:

x̄w = Σwx / Σw

**Worked example (weighted mean):**
Suppose prices of two items are 10 and 12, and quantities purchased are 2 and 3.

Σwx = 10×2 + 12×3 = 20 + 36 = 56
Σw = 2 + 3 = 5
Weighted mean price = 56 / 5 = 11.2

Thus, mean gives an overall representative value, and weighted mean is more accurate when values have different importance or quantities.

Question 14

Explain median and mode and show how each is useful in business. Include a small example.

Accepted Answer

Median and mode are positional averages that are especially useful when data is skewed or when a “most common” value is needed.

**Median** is the middle value of an ordered data set. It is not affected by extreme values, so it is useful for wages or income data where a few very high values can distort the mean.

**Example (median):**
Data: 5, 8, 10, 12, 15 (already arranged)
Median = 3rd item = 10

**Mode** is the value with the highest frequency. Mode is useful when a business wants to know the most common size, model, or choice.

**Example (mode):**
Shoe sizes sold: 7, 8, 8, 9, 8, 10
Mode = 8 (most frequently sold)

Thus, median represents the central value without being affected by outliers, and mode represents the most popular or typical value in business situations.

Question 15

Write a detailed note on selecting an appropriate measure of central tendency. Use a comparison table.

Accepted Answer

No single average is best in all cases. The choice of mean, median or mode depends on the nature of data and the purpose of analysis.

| Situation | Best measure | Reason |
|---|---|---|
| Data is symmetric and no extreme values | Mean | uses all values and supports further calculations |
| Data is skewed or has outliers (income, wages) | Median | not affected by extreme values |
| Need the most common value (popular size/brand) | Mode | shows most frequent/typical value |
| Qualitative/categorical emphasis | Mode | identifies most common category |

For example, average income using mean can be misleading if a few people have very high incomes. In such cases, median provides a better “typical” income. Similarly, a retailer chooses mode to know the most demanded size.

Therefore, selecting the correct measure of central tendency improves accuracy, interpretation and decision making in business.

Point	Mean	Median	Mode
Basis	arithmetic average	middle value	most frequent value
Uses	further calculations	skewed/outlier data	typical/most common
Affected by extremes	yes	no	usually no

Measures of Central Tendency: Mean, Median, Mode and Applications

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