Range, Quartile Deviation, Mean Deviation, or Standard Deviation?

An average can tell you where a group is centred. It cannot tell you how tightly the values stay around that centre.

Imagine two shops with average daily sales of 50 units.

Shop A: 48, 49, 50, 51, 52
Shop B: 10, 30, 50, 70, 90

Both means are 50, but the sales patterns are not remotely alike. Shop A is steady. Shop B swings sharply from one day to another.

This difference is called dispersion, or spread.

The confusing part is that we have several ways to measure it: range, quartile deviation, mean deviation, and standard deviation. They are not four decorative versions of the same formula. Each one looks at the data differently.

This guide will help you decide which measure fits, calculate each one correctly, and explain what your answer means.

Start With the Question, Not the Formula

Before calculating anything, ask what kind of spread you need to see.

What you want to know	Measure that usually fits
How wide is the complete span from smallest to largest?	Range
How spread out is the middle half of the data?	Quartile deviation
What is the average absolute distance from a chosen centre?	Mean deviation
How much overall variation is there around the mean, with larger gaps receiving more weight?	Standard deviation

This is the decision behind the formulas.

If a question asks about the full span of temperatures, range may be enough. If a few unusually high incomes would distort the picture, quartile deviation may describe the typical group better. If you want an easy-to-explain average distance, mean deviation is useful. If every value matters and larger deviations should count more strongly, standard deviation is usually the strongest choice.

1. Range: The Fastest View of the Full Span

Range uses only the largest and smallest values.

Range = Largest value - Smallest value

For the data:

12, 15, 18, 20, 27

the range is:

Range = 27 - 12 = 15

The answer tells us that the full data set stretches across 15 units.

When range is useful

Use range when:

• you need a quick first look at spread
• the extreme values are themselves important
• you are checking the total span of prices, temperatures, output, or marks
• the data set is small and a rough comparison is enough

For example, a shop owner checking the day’s highest and lowest delivery time may care directly about the total gap. Range answers that question immediately.

The weakness of range

Range ignores every value except two.

Compare these series:

Series A: 10, 11, 12, 13, 20
Series B: 10, 15, 15, 15, 20

Both have a range of 10, yet their inner values are arranged differently. Range cannot see that difference.

It is also highly sensitive to one extreme value. If the final value in Series A changes from 20 to 100, the range jumps from 10 to 90 even though four of the five values remain unchanged.

2. Quartile Deviation: The Spread of the Middle Half

Quartiles divide ordered data into four parts.

• Q1 marks the lower quarter.
• Q2 is the median.
• Q3 marks the upper quarter.

The distance from Q1 to Q3 is the interquartile range:

Interquartile Range = Q3 - Q1

Quartile deviation is half of that distance:

Quartile Deviation = (Q3 - Q1) / 2

It is therefore also called the semi-interquartile range.

Consider the ordered values:

4, 5, 5, 6, 6, 7, 23

Here:

Q1 = 5
Q3 = 7

So:

Interquartile Range = 7 - 5 = 2
Quartile Deviation = 2 / 2 = 1

The value 23 is far from the rest, but it does not change Q1 or Q3 in this example. Quartile deviation keeps its attention on the middle 50 percent.

When quartile deviation is useful

Use quartile deviation when:

• the median is a better centre than the mean
• the distribution is skewed
• there are unusually high or low observations
• you want the spread of the typical middle group
• a distribution has an open end and the extreme boundary is unavailable

Income and property-price data often contain a small number of very large values. In such cases, the middle half may tell a more useful story than the full range.

What quartile deviation leaves out

Its strength is also its limitation. Quartile deviation ignores the lowest quarter and the highest quarter of the observations. It is stable in the presence of extremes, but it cannot describe what is happening in the tails.

3. Mean Deviation: The Average Absolute Distance

Suppose we subtract the mean from every observation. Some deviations will be positive and some negative. If we add them normally, they cancel and the total becomes zero.

Mean deviation solves this by taking the absolute value of every deviation.

For ungrouped data, mean deviation about a central value A is:

Mean Deviation about A
= Sum of |x - A|
  / Number of observations

The centre A may be the mean or the median, depending on the question.

Take the data:

2, 4, 6, 8

The mean is 5.

x	x - 5	Absolute deviation
2	-3	3
4	-1	1
6	1	1
8	3	3

Therefore:

Mean Deviation about Mean
= (3 + 1 + 1 + 3) / 4
= 2

The values are, on average, 2 units away from the mean.

Mean deviation about mean or median?

The wording decides the centre.

M.D. about mean
-> deviations from the mean

M.D. about median
-> deviations from the median

Do not calculate deviations from whichever centre looks easier. Use the one named in the question.

Mean deviation about the median has a useful property: among all possible centres, the median gives the smallest total absolute deviation. That makes it a natural partner for skewed data.

When mean deviation is useful

Use mean deviation when:

• you want to use every observation
• you want a result that reads as an average distance
• absolute differences are easier to explain than squared differences
• the question specifies mean deviation about the mean or median

Its limitation

Absolute values are intuitive, but they are less convenient for later algebraic work. This is one reason standard deviation appears more often in advanced analysis.

4. Standard Deviation: Overall Spread Around the Mean

Standard deviation also uses every observation, but it handles positive and negative deviations by squaring them.

For a complete ungrouped series treated as the population in a school-level question:

Variance = Sum of (x - mean)^2 / N
Standard Deviation = Square root of Variance

The square performs two jobs:

1. It makes every deviation non-negative.
2. It gives greater influence to observations farther from the mean.

The square root then returns the answer to the original unit. If the data is measured in rupees, the standard deviation is also measured in rupees. Variance would be measured in square rupees, which is harder to interpret directly.

For the data:

2, 4, 6, 8

the mean is 5.

x	x - 5	(x - 5)^2
2	-3	9
4	-1	1
6	1	1
8	3	9

So:

Variance
= (9 + 1 + 1 + 9) / 4
= 5

Standard Deviation
= Square root of 5
= 2.24, approximately

Notice that mean deviation was 2, while standard deviation is about 2.24. Squaring gives the larger deviations more influence.

When standard deviation is useful

Use standard deviation when:

• every observation should affect the result
• the mean is the chosen centre
• large departures from the mean deserve extra attention
• you need a widely used measure for further statistical analysis
• you are comparing the spread of series with the same or very similar means

Population or sample formula?

Some later statistics courses distinguish between a complete population and a sample taken from a larger population. The population formula divides by N, while the sample formula commonly divides by n - 1.

In a school exercise, use the formula stated or implied by the chapter. Do not switch denominators halfway through a solution.

One Extreme Value, Four Different Reactions

The best way to understand the measures is to watch them react to the same change.

Consider:

Series A: 4, 5, 5, 6, 6, 7, 8
Series B: 4, 5, 5, 6, 6, 7, 23

Only the last value changes.

Measure	Series A	Series B	What changed?
Range	4	19	Changes sharply because it uses the maximum
Quartile deviation	1	1	Unchanged because the middle half is unchanged
Mean deviation about mean	1.02	4.29	Rises because every absolute distance is used
Standard deviation	1.25	6.19	Rises most strongly because large deviations are squared

The table gives us four different but valid descriptions.

• Range says the full span has become much wider.
• Quartile deviation says the middle group is just as tightly packed as before.
• Mean deviation says the average distance from the mean has increased.
• Standard deviation signals the distant value very strongly.

Which Measure Fits Which Situation?

Situation 1: Daily temperature span

You want the difference between the highest and lowest temperature recorded today.

Choose range. The extremes are exactly what you need.

Situation 2: Typical household income spread

A few households have exceptionally high incomes, and you want to describe the middle half.

Choose quartile deviation. It is less affected by the extreme tail.

Situation 3: Average distance from a target centre

You want to explain, in plain units, how far observations lie from the mean or median on average.

Choose mean deviation. Its absolute distances have a direct interpretation.

Situation 4: Overall variation in production

Every day’s output matters, and very large departures from average output should receive more weight.

Choose standard deviation. It includes all observations and emphasises larger gaps.

Situation 5: Comparing two series with different means

Suppose one shop averages sales of 20 units and another averages 200 units. Their standard deviations cannot be judged fairly without considering the scale of their means.

In that case, use a relative measure such as coefficient of variation. You can revise the full method in our guide to comparing consistency with coefficient of variation.

Absolute Measures and Relative Measures

Range, quartile deviation, mean deviation, and standard deviation are absolute measures. Their answers carry the unit of the original data.

If wages are in rupees:

• range is in rupees
• quartile deviation is in rupees
• mean deviation is in rupees
• standard deviation is in rupees

This makes them useful for describing one series. It may make comparisons unfair when two series use different units or have very different average sizes.

Relative measures remove the unit by forming a ratio or percentage. Examples include coefficient of range, coefficient of quartile deviation, coefficient of mean deviation, and coefficient of variation.

What Happens When Every Value Changes?

These two rules save time and help you check answers.

Adding or subtracting a constant

Suppose 5 is added to every observation. The whole series shifts upward, but the distances between values do not change.

Therefore, these remain unchanged:

• range
• quartile deviation
• mean deviation
• standard deviation

For example:

Original: 2, 4, 6, 8
New:      7, 9, 11, 13

Both series have range 6, mean deviation 2, and standard deviation about 2.24.

Multiplying or dividing by a constant

If every value is multiplied by 3, all distances become three times as large.

Therefore:

• range becomes 3 times the original range
• quartile deviation becomes 3 times the original quartile deviation
• mean deviation becomes 3 times the original mean deviation
• standard deviation becomes 3 times the original standard deviation

More generally, multiplying every value by k multiplies each of these measures by |k|.

The absolute value matters because a measure of spread cannot be negative.

How the Formulas Change for Frequency Data

When a value occurs several times, frequency tells us how much weight it receives.

Let:

x = observation or class midpoint
f = frequency
N = sum of frequencies
A = chosen centre

Then:

Mean Deviation about A = Sum of f|x - A| / N

and:

Standard Deviation
= Square root of
  [Sum of f(x - mean)^2 / N]

For a continuous distribution, use the class midpoint as x when calculating mean deviation or standard deviation.

Quartile deviation follows a different route. You first use cumulative frequency to locate Q1 and Q3, then apply:

Quartile Deviation = (Q3 - Q1) / 2

Range normally uses the extreme values or class boundaries. Be careful with open-ended classes because the true smallest or largest boundary may be unknown.

A Reliable Decision Path

Use this sequence in any selection question:

1. Read the wording. Does it ask for total span, middle spread, average absolute distance, or overall variation around the mean?
2. Look for extremes. Are unusually high or low values important, or are they likely to distort the typical picture?
3. Identify the centre. Is the question built around the median or the mean?
4. Check how much data should be used. Two extremes, the middle half, or every observation?
5. Check whether it is one series or a comparison. Different means or units may require a relative measure.
6. Write one line of interpretation. A calculation without meaning is incomplete.

For example:

Standard deviation = 4.6 marks

is stronger when followed by:

The marks show an overall spread of about 4.6 marks around the mean, with larger deviations receiving more weight.

Common Mistakes to Avoid

1. Finding range without ordering or checking the extremes

Range needs the true largest and smallest values. Do not use the first and last values unless the data is already arranged.

2. Confusing interquartile range with quartile deviation

IQR = Q3 - Q1
QD  = (Q3 - Q1) / 2

Quartile deviation is half the interquartile range.

3. Forgetting absolute signs in mean deviation

If you add ordinary deviations from the mean, positive and negative values cancel. Mean deviation requires |x - A|.

4. Using the wrong centre for mean deviation

Mean deviation about median must use the median. Mean deviation about mean must use the mean.

5. Forgetting to square deviations for variance

Standard deviation comes from squared deviations, not absolute deviations.

6. Stopping at variance

If the question asks for standard deviation, take the positive square root of variance.

7. Ignoring frequencies

In a frequency distribution, multiply each distance or squared distance by its frequency.

8. Treating a larger standard deviation as automatically bad

A larger standard deviation means greater spread. Whether that is desirable depends on the context.

9. Comparing absolute measures across very different means

A standard deviation of 10 may be large around a mean of 20 and small around a mean of 500. Use a suitable relative measure when scale matters.

10. Writing only the number

Always attach the unit and a short interpretation.

Practice Questions

Try these before checking the answers.

1. Find the range of 7, 12, 15, 18, 21.
2. For 4, 5, 5, 6, 6, 7, 23, find the interquartile range and quartile deviation using Q1 = 5 and Q3 = 7.
3. Find mean deviation about the mean for 2, 4, 6, 8.
4. Find the population standard deviation of 2, 4, 6, 8 to two decimal places.
5. Every value in a series is increased by 20. What happens to its standard deviation?
6. Every value in a series is multiplied by 4. What happens to its quartile deviation?
7. One salary is extremely high compared with the rest. Which measure best describes the spread of the middle half?
8. Two series have the same mean. Their standard deviations are 3 and 7. Which is more consistent?
9. Why can range miss an important difference between two distributions?
10. Which measure uses all observations but does not square their distances from the centre?

Answers

1. 21 - 7 = 14.
2. IQR = 7 - 5 = 2; QD = 2 / 2 = 1.
3. The mean is 5, so M.D. = (3 + 1 + 1 + 3) / 4 = 2.
4. Variance is 5, so standard deviation is sqrt(5) = 2.24, approximately.
5. It remains unchanged because adding a constant changes the centre but not the distances.
6. It becomes four times the original quartile deviation.
7. Quartile deviation.
8. The series with standard deviation 3 is more consistent because the means are equal.
9. Range uses only the largest and smallest observations, so it cannot see how the inner values are arranged.
10. Mean deviation.

Sources and Further Reading

• NIOS: Measures of Dispersion
• NIOS Economics: Measures of Dispersion overview
• OpenStax: Measures of Variation
• OpenStax: Quartiles, Interquartile Range, and Potential Outliers
• NIST: Why Different Measures of Scale Are Useful

Frequently Asked Questions

What is the simplest measure of dispersion?

Range is the simplest. Subtract the smallest value from the largest value.

Which measure of dispersion is best?

There is no single best measure for every purpose. Range fits a quick span, quartile deviation fits the middle half, mean deviation gives an average absolute distance, and standard deviation gives a fuller measure around the mean.

What is the difference between interquartile range and quartile deviation?

Interquartile range is Q3 - Q1. Quartile deviation is half of it, so QD = (Q3 - Q1) / 2.

Why is quartile deviation less affected by extreme values?

It is based on Q1 and Q3, so it focuses on the middle 50 percent and leaves out the outer quarters.

Can quartile deviation be zero?

Yes. If Q1 and Q3 are equal, then their difference is zero and quartile deviation is zero, even if a few values outside the middle half differ.

Why do we use absolute values in mean deviation?

Ordinary deviations above and below the centre have opposite signs and cancel. Absolute values preserve the size of every distance.

Is mean deviation always calculated from the mean?

No. It may be calculated about the mean or the median. The question should tell you which centre to use.

Why are deviations squared in standard deviation?

Squaring prevents positive and negative deviations from cancelling and gives more influence to observations far from the mean.

Can standard deviation be negative?

No. Variance is non-negative, and standard deviation is its positive square root. It can be zero only when all observations are equal.

Does adding the same number to every value change dispersion?

It shifts the whole series but does not change the distances. Range, quartile deviation, mean deviation, and standard deviation remain unchanged.

Does multiplying every value change dispersion?

Yes. Multiplying every value by k multiplies each absolute measure of dispersion by |k|.

When should I use coefficient of variation instead of standard deviation?

Use coefficient of variation when comparing relative spread across series with different means or scales. Standard deviation alone is an absolute measure and may not give a fair comparison in that situation.