Mean, Median and Mode: Positive and Negative Skewness

Imagine asking seven shopkeepers how much they spent on a delivery. Most paid a modest amount. One paid much more because the parcel travelled a long distance.

If you calculate the mean, that expensive delivery gets a say in the answer. If you find the median, you look at the middle delivery after arranging the charges. If you find the mode, you look for the charge repeated most often.

Same deliveries. Three different questions. That is why the answers can differ without any calculation being wrong.

The relationship between mean, median and mode helps us read the shape of a distribution. Let’s make the ordering understandable, practise the formula, and learn when a shortcut needs a little caution.

The quick answer: how are mean, median and mode related?

For the familiar single-peaked distributions used in introductory exercises, remember this pattern:

Shape	Usual order, smallest to largest	Where the longer tail points
Positively skewed	Mode < Median < Mean	Right, towards higher values
Negatively skewed	Mean < Median < Mode	Left, towards lower values
Symmetric with one central peak	Mean = Median = Mode	Neither side dominates

The ordering for skewed distributions is a useful usual pattern, not a rule that every possible dataset must obey. OpenStax’s discussion of skewness makes this qualification explicit, particularly for discrete data.

For a moderately skewed distribution, the familiar empirical relationship is:

Mode ≈ 3 × Median - 2 × Mean

The symbol ≈ means “approximately equal to”. If an exercise explicitly tells you to use the empirical relationship, use the formula to obtain the required estimate.

First, understand the three centres

The three measures are easier to remember when you attach a question to each one.

Mean: What would each observation receive if the total were shared equally? Add all the values and divide by their number. For a frequency table, use Mean = Σfx / Σf, so a value occurring ten times contributes ten times.

Median: What sits in the middle of the ordered observations? Arrange individual values first. For an odd number of observations, take the middle one. For an even number, average the two middle values.

Mode: Which value occurs most often? Count repetitions. A dataset can have one mode, several modes, or no single most frequent value.

These are different ways to describe location, as explained in the NIST handbook’s Measures of Location. The mean uses every numerical value; the median uses position; the mode uses frequency.

Think of houses along a road. The mode is the busiest address value, the median is the halfway observation in an ordered walk, and the mean is the balance point if every observation has equal weight. A distant observation matters greatly to balance even if few observations live near it.

Positive skewness: follow the tail to the right

Suppose the delivery charges, in rupees, are:

20, 20, 20, 30, 40, 50, 100

Most charges are near the lower end. The Rs 100 delivery stretches the distribution towards higher values.

Calculate the mean

Total = 20 + 20 + 20 + 30 + 40 + 50 + 100 = Rs 280.

Mean = 280 / 7 = Rs 40.

Find the median

There are seven observations, already in ascending order. The middle position is (7 + 1) / 2 = 4.

The fourth observation is Rs 30.

Find the mode

Rs 20 occurs three times. Every other charge occurs once.

Mode = Rs 20.

Our order is therefore:

Mode 20 < Median 30 < Mean 40.

Notice something useful: four of the seven charges are below the mean. There is no rule saying that the mean must divide the observations into two equal groups. That is the median’s job, with allowances for ties.

This is a clear right-tail example. “Positive” does not mean that every observation is positive or that the outcome is good. It describes the direction of asymmetry.

Negative skewness: follow the tail to the left

Now consider seven scores out of 100:

0, 50, 60, 70, 80, 80, 80

Most scores are high. One very low score stretches the distribution towards the left.

Total = 0 + 50 + 60 + 70 + 80 + 80 + 80 = 420.

Mean = 420 / 7 = 60 marks.

The fourth observation is 70 marks, so the median is 70.

The most frequent score is 80 marks, so the mode is 80.

Now the order is:

Mean 60 < Median 70 < Mode 80.

Compare this with the delivery example. The unusual value has moved from the high side to the low side, and the mean has followed it.

Do not write that negative skewness means negative observations. All the scores here are non-negative. A low tail is enough to illustrate the idea.

How to read or draw the skewness curve

On the usual frequency graph, values increase along the horizontal axis and frequency increases upwards.

For a right-skewed curve, draw one peak nearer the lower-value side, then let the curve taper gradually to the right. In the standard sketch, mark the mode below the peak, the median farther right, and the mean farther right again.

For a left-skewed curve, reverse that picture. The thin tail stretches left, and the usual left-to-right labels are mean, median, mode.

For a symmetric, single-peaked curve, draw mirror-image sides and put all three labels at the central peak. Equal spacing between the labels is not required in a skewed sketch.

NIST’s guide to skewed histograms identifies the tail direction as the key visual feature. A high peak tells you where observations concentrate; a long tail tells you which direction stretches out.

This is a reading aid for the usual sketches, not a substitute for inspecting awkward or irregular data carefully.

The empirical relationship: where the formula comes from

For moderately skewed distributions, the empirical relationship is often introduced as:

Mean - Mode ≈ 3 × (Mean - Median)

Expanding the brackets gives:

Mean - Mode ≈ 3 × Mean - 3 × Median.

Rearranging gives:

Mode ≈ 3 × Median - 2 × Mean.

This is an observed approximation, not an algebraic identity that follows from the definitions of the three averages. The University of Kalyani’s study material on measures of central tendency, page 178, presents it for moderately skewed distributions.

You can rearrange it according to what is missing:

Required estimate	Formula
Mode	3 × Median - 2 × Mean
Median	(Mode + 2 × Mean) / 3
Mean	(3 × Median - Mode) / 2

The brackets matter. In the median formula, divide the entire numerator by 3. In the mean formula, divide the entire numerator by 2.

Example 1: estimate the mode

A moderately skewed distribution of weekly travel spending has mean Rs 260 and median Rs 240. Estimate the mode.

Mode ≈ 3 × 240 - 2 × 260

= 720 - 520

= Rs 200.

The order 200 < 240 < 260 matches the usual positive-skew pattern.

A good final sentence is: “The estimated mode is Rs 200, using the empirical relationship.”

Example 2: estimate the median

Mean sales are 48 units and modal sales are 42 units. Use the empirical relationship to estimate median sales.

Median ≈ (42 + 2 × 48) / 3

= (42 + 96) / 3

= 138 / 3

= 46 units.

Check: 3 × 46 - 2 × 48 = 138 - 96 = 42. Substitution recovers the supplied mode.

Example 3: estimate the mean

Median marks are 68 and modal marks are 74. Estimate the mean using the empirical relationship.

Mean ≈ (3 × 68 - 74) / 2

= (204 - 74) / 2

= 130 / 2

= 65 marks.

The estimated mean is below the median, which is below the mode. That fits the usual negative-skew pattern.

Why the formula does not always reproduce the actual mode

Return to our delivery charges:

20, 20, 20, 30, 40, 50, 100.

We calculated mean = 40, median = 30 and actual mode = 20.

The empirical formula would give:

Estimated mode = 3 × 30 - 2 × 40 = 10.

But 10 is not even an observed delivery charge. The actual mode remains 20.

Nothing has gone wrong with the arithmetic. The small, strongly uneven dataset simply does not satisfy the empirical relationship well.

This is why you should not calculate a mode from the formula when the raw observations already let you identify it directly. An estimate cannot overrule the frequency count.

Here is a second caution about ordering. Take:

0, 0, 1, 2, 3, 4, 4, 4.

Mean = 18 / 8 = 2.25.

Median = (2 + 3) / 2 = 2.5.

Mode = 4.

The values happen to follow mean < median < mode, but a proper description of shape should still consider the full distribution. Three summaries compress information; they do not show every bump and gap.

For an example that breaks the usual “median in the middle” ordering altogether, use:

0, 0, 0, 10, 11.

Mean = 21 / 5 = 4.2, while both median and mode are 0. The right-hand observations pull the mean up, but the median is not strictly between the other two measures.

The sensible habit is to use the standard pattern for standard questions and the actual observations for actual calculations.

Symmetry does not always mean all three are equal

Consider:

2, 3, 4, 4, 4, 5, 6.

The values balance around 4. Their total is 28, so the mean is 4. The fourth value is 4, and 4 is the most frequent value.

Here, Mean = Median = Mode = 4.

Now consider another symmetric set:

1, 1, 2, 3, 4, 4.

Mean = 15 / 6 = 2.5.

Median = (2 + 3) / 2 = 2.5.

However, there are two modes: 1 and 4.

Symmetry has not produced a unique central mode. That is why “symmetric with one central peak” is a more careful statement than simply “symmetric” when claiming equality of all three averages.

Which average should a commerce student report?

Suppose a small business records daily customer spending. The owner may want the mean to connect total spending with customer count. A student describing an ordinary customer’s experience may also report the median if a few large orders dominate the total. The most frequent purchase amount answers a different question again.

Try this miniature example:

Rs 100, Rs 100, Rs 120, Rs 140, Rs 540.

Mean = 1,000 / 5 = Rs 200.

Median = Rs 120.

Mode = Rs 100.

If the last purchase increases from Rs 540 to Rs 1,040, the total rises by Rs 500. With five customers, the mean increases by 500 / 5 = Rs 100, becoming Rs 300.

The median stays Rs 120 and the mode stays Rs 100. Their answers are unchanged because the middle observation and the most frequent amount have not changed.

That does not make the mean useless. It now correctly reflects the higher total. It does show why the statement “the typical customer spent Rs 300” could mislead a reader who expects a middle customer’s bill.

For calculations involving class intervals, use the appropriate grouped-data method. Our guide to finding the median and modal class works through that separate task. When frequencies or importance weights are involved, the guide to choosing the correct weights for an arithmetic mean helps prevent denominator mistakes.

Common mistakes and quick repairs

Mistake	Better approach
Calling a curve positive because its peak is on the right	Look at the tail, not the peak
Treating negative skewness as evidence of negative values	Describe the direction of the lower-value tail
Writing Mode = 2 × Median - 3 × Mean	Start from 3 × Median = Mode + 2 × Mean
Using the empirical formula despite having exact raw data	Calculate the actual measures directly
Saying the median is always unaffected by changing data	Check whether the middle position or values changed
Assuming all symmetric datasets have one mode	Check for two peaks, flat tops or ties
Calling a mean above the median absolute proof of the full shape	Treat it as a clue and inspect the distribution

Try these six questions

Work them out before reading the answers. For Questions 1 to 3, assume the empirical relationship is appropriate.

1. Mean expenditure is Rs 75 and median expenditure is Rs 70. Estimate the mode.
2. Mean output is 36 units and mode is 30 units. Estimate the median.
3. Median marks are 54 and mode is 60. Estimate the mean.
4. Find all three averages for 2, 2, 2, 3, 4, 5, 10. Which usual skewness pattern do they illustrate?
5. Find the mean, median and mode or modes of 2, 2, 3, 4, 5, 5. Is equality of all three justified?
6. A distribution has most observations near high values and a thin tail extending towards low values. Name its skewness and give the usual order of the averages.

Answers with working

1. Mode ≈ Rs 60. Calculate 3 × 70 - 2 × 75 = 210 - 150 = 60.

2. Median ≈ 34 units. Calculate (30 + 2 × 36) / 3 = 102 / 3 = 34.

3. Mean ≈ 51 marks. Calculate (3 × 54 - 60) / 2 = 102 / 2 = 51.

4. Mean = 4, median = 3, mode = 2. The sum is 28 and there are seven observations. The fourth observation is 3, while 2 appears three times. The data illustrate the usual positive-skew pattern.

5. Mean = 3.5, median = 3.5, modes = 2 and 5. The sum is 21. The middle two observations are 3 and 4. The data are symmetric around 3.5 but have two modes, so the three measures do not coincide.

6. Negative skewness. The tail points left, towards lower values. The usual single-peaked pattern is Mean < Median < Mode.

Frequently asked questions

What is positive skewness in simple words?

It is asymmetry with a tail extending towards higher values on the right. Many observations may be near the lower end, with relatively few high values stretching out the tail.

What is negative skewness in simple words?

It is asymmetry with a tail extending towards lower values on the left. It does not require any observation to be below zero.

What is the usual order of mean, median and mode in positive skewness?

For the standard single-peaked example, Mode < Median < Mean. Treat this as a usual pattern rather than a universal rule for every dataset.

What is the usual order in negative skewness?

For the standard single-peaked example, Mean < Median < Mode. The mean lies towards the lower-value tail.

What is the empirical formula connecting the three averages?

Mode ≈ 3 × Median - 2 × Mean. Equivalently, 3 × Median ≈ Mode + 2 × Mean. It is an approximation associated with moderately skewed distributions.

Can I find the exact mode from the mean and median alone?

Not generally. You can estimate it when the empirical relationship is appropriate or explicitly requested. Exact raw-data frequencies, when available, determine the actual mode.

Are mean, median and mode equal in every symmetric distribution?

No. They coincide in the familiar symmetric distribution with one central peak, provided the mean exists. A symmetric dataset can have two modes away from its centre.

Is the median always better than the mean?

No. The median is useful for describing the middle when extreme observations distort the impression given by the mean. The mean is useful when every numerical value and the relationship with the total matter. Choose according to the question.

Does a mean greater than the median prove positive skewness?

It suggests the usual positive-skew pattern, but it does not fully determine the shape or every possible measure of skewness. Inspect the graph or data when available.

How should I present an empirical-relationship answer?

Write the formula, substitute both given values, show the arithmetic, and include the unit. Call the result an estimate so the reader knows how you obtained it.