Measures of dispersion range

28/June/2025 01:53 Share:

Here is a detailed explanation of Measures of Dispersion, broken into clear sections to cover all your topics:

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What is Dispersion?

Dispersion refers to the degree to which individual values in a dataset deviate or spread out from the average (mean, median, or mode). While measures of central tendency give a single representative value, measures of dispersion help understand the variability or consistency of the data.

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Objectives and Importance of Dispersion

1. To understand variability in data.

2. To compare consistency of two or more datasets.

3. To supplement averages for better data interpretation.

4. To support decision-making in economics, business, and management.

5. To measure reliability of statistical conclusions.

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Types of Dispersion (Measures of Dispersion)

1. Range

2. Interquartile Range

3. Mean Deviation

4. Standard Deviation

5. Coefficient of Variation

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1. Range

Meaning:

The range is the difference between the largest and smallest values in a dataset.

\text{Range} = \text{Largest Value} - \text{Smallest Value}

Example:

Data: 10, 15, 20, 25, 30

Range = 30 – 10 = 20

Merits:

Very simple to compute.

Gives a quick idea of spread.

Demerits:

Based only on extreme values.

Highly sensitive to outliers.

Not reliable for large datasets.

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2. Interquartile Range & Semi-Interquartile Range

Interquartile Range (IQR):

IQR = Q_3 - Q_1

Semi-Interquartile Range:

\frac{Q_3 - Q_1}{2}

Merits:

Not affected by extreme values.

Focuses on middle 50% of the data.

Demerits:

Ignores 50% of data.

Not suitable for mathematical treatment.

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3. Mean Deviation (Average Deviation)

Meaning:

It is the average of the absolute deviations from a central value (mean, median, or mode).

\text{Mean Deviation (from Mean)} = \frac{\sum |X - \bar{X}|}{n}

Merits:

Considers all values.

Less sensitive to extreme values than standard deviation.

Demerits:

Ignores the signs of deviation.

Not useful for algebraic calculations.

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4. Standard Deviation (SD)

Meaning:

It is the square root of the average of the squared deviations from the mean.

\sigma = \sqrt{\frac{\sum (X - \bar{X})^2}{n}}

It is the most reliable and widely used measure of dispersion.

Methods of Calculating SD:

1. Direct Method

2. Assumed Mean Method

3. Step-Deviation Method

Each method involves finding the deviation of each value from the mean and then squaring, summing, and averaging them.

Merits:

Based on all observations.

Suitable for algebraic operations.

Most precise measure of dispersion.

Demerits:

Complicated calculations.

Sensitive to extreme values.

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5. Coefficient of Variation (CV)

Meaning:

It is the ratio of standard deviation to the mean, expressed as a percentage.

CV = \left( \frac{\sigma}{\bar{X}} \right) \times 100

Use:

To compare the variability of two or more datasets regardless of units.

Lower CV indicates higher consistency.

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Comparison of Different Measures of Dispersion

Measure Considers All Values Sensitive to Outliers Suitable for Further Analysis Best Used When

Range No Yes No Quick estimates

Interquartile Range No No No Non-normal data

Mean Deviation Yes Medium No (due to modulus) Moderate dispersion

Standard Deviation Yes Yes Yes All purposes

Coefficient of Variation Yes Yes Yes Comparing datasets

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Conclusion

Dispersion measures are essential to understand the consistency, reliability, and spread of a dataset. While mean gives you the central value, dispersion tells how scattered the data is. Among all, standard deviation is the most popular due to its algebraic strength, and coefficient of variation is excellent for comparison across different units or scales.