Measures of Central Value | ISC Class 11 Economics Notes

This note covers the meaning of a measure of central value, the arithmetic mean (simple and weighted) for ungrouped and grouped data, the median for ungrouped and grouped data, the mode for ungrouped and grouped data, and the relationship between mean, median and mode for symmetrical, positively skewed and negatively skewed frequency distributions.

What is a measure of central value?

A set of raw data such as the incomes of fifty families, or the marks of forty students, is too large to describe item by item.

A measure of central value (also called an average) summarises the whole data set in one single number that is a typical or representative value of the entire distribution.

The three most commonly used measures of central value are the arithmetic mean, the median and the mode.

Each of these looks at the data differently: the mean uses every value in the series, the median looks at the middle position when the data is arranged in order, and the mode looks at the value that occurs most often.

Choosing which average to use depends on the purpose of the analysis and the nature of the data.

For example, the average size of a readymade garment is best described by the mode (the size most people buy), while the average wage paid in a firm is best described by the mean (because it uses every worker's wage).

An Indian example used to introduce this idea is that of a farmer named Baiju, who grows food grains on his one-acre plot in an Indian village called Balapur in Buxar district of Bihar. Balapur has 50 small farmers, and to judge Baiju's economic condition one must compare his land holding with the holdings of all the other farmers in the village, not with any single farmer's holding.

Three different averages give three different answers to the same question about Baiju's land in this Indian village: whether his holding is above average in the ordinary sense (the arithmetic mean of all holdings in Balapur), above the size that half the farmers own (the median), or above what most farmers own (the mode). This Indian farming example shows that a single raw figure is not enough; the whole distribution of land holdings in the village must first be summarised by a measure of central tendency before Baiju's position within it can be judged.

The weekly family incomes and factory daily wages used as worked examples through this topic are likewise expressed in Indian rupees (Rs), alongside examination marks and housing-plot sizes, reflecting the kind of household, workplace and school data that a measure of central tendency is used to summarise in an Indian context.

Definition: Measure of central value. A single value that represents the whole of a set of data by indicating the point about which the items tend to cluster.

What is the arithmetic mean and how is it calculated for ungrouped data?

The arithmetic mean, usually denoted X̄ ("X-bar"), is defined as the sum of the values of all observations divided by the number of observations. If there are N observations X₁, X₂, X₃, … , Xₙ, then:

Arithmetic Mean = ΣX/N, where ΣX is the sum of all observations and N is the total number of observations.

Direct method

By the direct method, the mean is simply the sum of all the observations divided by their number.

Worked example 1. The marks of five students in an economics test are 40, 50, 55, 78 and 58. Calculate the arithmetic mean.

Answer: X̄ = ΣX/N = (40 + 50 + 55 + 78 + 58)/5 = 281/5 = 56.2. The average mark of the students in the economics test is 56.2.

Assumed mean method and step deviation method

When the data has many observations and/or large figures, computing the mean directly becomes tedious.

The assumed mean method takes any value in the data (A) as an assumed mean, finds the deviation d = X − A of each observation from A, sums these deviations (Σd), and then adds the ratio Σd/N to A to get the true mean.

Symbolically, mean = A + Σd/N. Any value, whether it occurs in the data or not, can be used as the assumed mean, though a centrally located value simplifies the arithmetic.

The step deviation method divides each deviation by a common factor c, so that d′ = (X − A)/c, which reduces the size of the numbers further. The mean is then A + (Σd′/N) × c.

Worked example 2. The weekly incomes (in Rs) of ten families A to J are 850, 700, 100, 750, 5000, 80, 420, 2500, 400 and 360. Compute the mean family income by the assumed mean method, taking A = 850.

Answer: Deviations d = X − 850 are 0, −150, −750, −100, +4150, −770, −430, +1650, −450, −490, and Σd = +2660. Mean = A + Σd/N = 850 + 2660/10 = 850 + 266 = Rs 1,116.

Using the step deviation method with c = 10, Σd′ = 266, so mean = 850 + (266/10) × 10 = Rs 1,116, the same answer as before and as the direct method.

The working for worked example 2 is set out below as a table, showing each family's income, its deviation from the assumed mean of Rs 850, and the step-deviation value after dividing by the common factor c = 10.

How is the arithmetic mean calculated for grouped data?

For grouped data, each value is multiplied by its frequency (f) before being summed. The direct-method formula becomes:

Mean (grouped) = ΣfX/Σf, where ΣfX is the sum of the products of each value (or mid-value, for continuous series) and its frequency, and Σf is the total frequency.

Discrete series

Worked example 3. Plots in a housing colony come in three sizes, 100, 200 and 300 square metres, with 200, 50 and 10 plots of each size respectively. Find the mean plot size.

Answer: fX values are 100×200 = 20000, 200×50 = 10000 and 300×10 = 3000, so ΣfX = 33,000 and Σf = 260. Mean = ΣfX/Σf = 33000/260 = 126.92 square metres.

Continuous series

In a continuous series, data is given in class intervals, so the mid-point of each class is taken as the representative value before applying the discrete-series formula.

Class intervals may be exclusive (0 to 10, 10 to 20, …), inclusive (0 to 9, 10 to 19, …) or of unequal size, but the method of calculation is the same in each case.

Worked example 4. The marks of 70 students are distributed as follows: 5 students score 0 to 10, 12 score 10 to 20, 15 score 20 to 30, 25 score 30 to 40, 8 score 40 to 50, 3 score 50 to 60 and 2 score 60 to 70. Find the average mark by the direct method.

Answer: Mid-values are 5, 15, 25, 35, 45, 55, 65. Products fm are 25, 180, 375, 875, 360, 165 and 130, so Σfm = 2,110 and Σf = 70. Mean = Σfm/Σf = 2110/70 = 30.14 marks.

Two properties of the arithmetic mean follow from its definition: (i) the sum of the deviations of all items from the mean is always zero, written Σ(X − X̄) = 0, and (ii) the mean is affected by extreme values, because any unusually large or small value pulls the whole sum up or down.

What is the weighted arithmetic mean?

A simple arithmetic mean treats every observation as equally important. When some items matter more than others, each value X is multiplied by a weight W reflecting its importance, and the weighted arithmetic mean is used instead.

Weighted Mean = ΣWX/ΣW, where W is the weight assigned to each value X according to its relative importance.

A weighted mean is appropriate, for instance, when comparing the cost of a basket of goods such as mangoes and potatoes, where the quantity of each good purchased acts as its weight, so that a commodity consumed in larger quantity has a proportionately greater influence on the average price.

This matters because a simple (unweighted) mean of the prices of mangoes and potatoes would give the two commodities equal importance, even though a household may buy far more potatoes than mangoes in a week. By multiplying each price by the quantity bought before averaging, the weighted mean reflects the actual importance of each commodity in the household's overall spending, rather than treating every commodity as if it mattered equally.

The choice of weight is therefore a matter of judgement about what makes one item more important than another in the context of the problem: quantity purchased, quantity produced, number of workers, or any other measure of relative significance appropriate to the data being averaged.

The weighted mean is calculated in the same broad way as the mean for grouped data, except that the weight W plays the role that the frequency f plays in the grouped-data formula, so that ΣWX replaces ΣfX and ΣW replaces Σf in the working.

Worked example 5. A household buys 4 kg of potatoes at Rs 20 per kg and 1 kg of mangoes at Rs 100 per kg. Find the weighted mean price per kg, using the quantity bought as the weight.

Answer: ΣWX = (4 × 20) + (1 × 100) = 80 + 100 = 180, and ΣW = 4 + 1 = 5. Weighted mean price = ΣWX/ΣW = 180/5 = Rs 36 per kg.

The simple mean of the two prices would be (20 + 100)/2 = Rs 60 per kg, which overstates the average price actually paid, because far more potatoes than mangoes were bought.

What is the median and how is it calculated for ungrouped data?

The median is the value of the middle item when all observations are arranged in ascending or descending order, so that an equal number of observations lie above and below it. Its position in an ordered series of N items is given by:

Median position = (N + 1)/2, that is, the median is the (N + 1)/2th item of the ordered array.

When N is odd, the median is simply the value of this middle item. When N is even, there are two middle observations, and the median is the arithmetic mean of the two.

Worked example 6. The marks of 20 students are 25, 72, 28, 65, 29, 60, 30, 54, 32, 53, 33, 52, 35, 51, 42, 48, 45, 47, 46, 33. Find the median mark.

Answer: Arranged in ascending order: 25, 28, 29, 30, 32, 33, 33, 35, 42, 45, 46, 47, 48, 51, 52, 53, 54, 60, 65, 72. With N = 20, the two middle observations are the 10th and 11th values, 45 and 46. Median = (45 + 46)/2 = 45.5 marks.

How is the median calculated for grouped data?

Discrete series

For a discrete series, the median's position (N + 1)/2 is located through the cumulative frequency column, and the value corresponding to that position is the median.

Worked example 7. Twenty persons have incomes of Rs 10, 20, 30 and 40, with 2, 4, 10 and 4 persons respectively earning these incomes. Find the median income.

Answer: Cumulative frequencies are 2, 6, 16, 20. The median lies at the (N + 1)/2 = 21/2 = 10.5th observation, which falls in the cumulative frequency of 16 (income Rs 30). The median income is Rs 30.

Continuous series

For continuous series, the median class is the class in which the N/2th item (not the (N+1)/2th item) lies. The median is then:

Median = L + ((N/2 − cf)/f) × h, where L is the lower limit of the median class, cf is the cumulative frequency of the class preceding the median class, f is the frequency of the median class, and h is the class width (magnitude) of the median class interval.

Worked example 8. Daily wages (in Rs) of 160 factory workers are distributed as: 20 to 25 → 14 workers, 25 to 30 → 28, 30 to 35 → 33, 35 to 40 → 30, 40 to 45 → 20, 45 to 50 → 15, 50 to 55 → 13, 55 to 60 → 7. Find the median daily wage.

Answer: Cumulative frequencies rise to 14, 42, 75, 105, 125, 140, 153, 160. With N/2 = 80, the median class is 35 to 40, where L = 35, cf (of the preceding class, 30 to 35) = 75, f = 30, h = 5.

Median = 35 + ((80 − 75)/30) × 5 = 35 + 0.83 = Rs 35.83. This means that 50% of the workers earn Rs 35.83 or less, and 50% earn Rs 35.83 or more.

The median is not sensitive to every value in the series; it concentrates only on the values of the central items, so it is not pulled about by extreme values the way the mean is.

What is the mode and how is it calculated?

The mode, denoted Mo, is the value that is repeated the highest number of times in a series, that is, the most frequently observed data value. The term comes from the French "la Mode", meaning the most fashionable or common value.

In a series such as 1, 2, 3, 4, 4, 5, the mode is 4 because it occurs most often.

A distribution with one such value is unimodal; one with two most-frequent values is bimodal; and one with more than two is multi-modal.

A series such as 1, 1, 2, 2, 3, 3, 4, 4 has no mode at all, because no single value appears more often than the others.

Mode for discrete series

For a discrete frequency distribution, the mode is found by inspection: the value of the variable against which the frequency is highest is the mode.

Worked example 9. A discrete series gives the variable values 10, 20, 30, 40 and 50 with frequencies 2, 8, 20, 10 and 5 respectively. Find the mode.

Answer: The maximum frequency is 20, and it belongs to the variable value 30. The mode is therefore 30.

As there is only one such value, the data is unimodal.

Mode for continuous series

For a continuous frequency distribution, the modal class is the class with the largest frequency. The mode is then:

Mode = L + (D₁/(D₁ + D₂)) × h, where L is the lower limit of the modal class, D₁ is the difference (ignoring signs) between the frequency of the modal class and that of the class preceding it, D₂ is the difference (ignoring signs) between the frequency of the modal class and that of the class succeeding it, and h is the class interval width.

The class intervals must be equal in size and the series exclusive for this formula to be used.

Worked example 10. A cumulative frequency table of monthly family income (in '000 Rs) gives "less than" totals: less than 15 → 4, less than 20 → 12, less than 25 → 30, less than 30 → 60, less than 35 → 80, less than 40 → 90, less than 45 → 95, less than 50 → 97. Find the modal income.

Answer: Converting to an ordinary frequency table by subtraction gives class frequencies of 4 (10 to 15), 8 (15 to 20), 18 (20 to 25), 30 (25 to 30), 20 (30 to 35), 10 (35 to 40), 5 (40 to 45) and 2 (45 to 50).

The class with the largest frequency, 30, is 25 to 30, so this is the modal class, with L = 25, D₁ = (30 − 18) = 12 (the preceding class, 20 to 25, has frequency 18), D₂ = (30 − 20) = 10 (the succeeding class, 30 to 35, has frequency 20), h = 5.

Mode = 25 + (12/(12 + 10)) × 5 = 25 + 2.727 = 27.727 (in '000 Rs). The modal worker family's monthly income is therefore about Rs 27,727.

What is the relationship between mean, median and mode?

If the arithmetic mean is written Me, the median Mi and the mode Mo, their relative sizes follow one of two orders: Me > Mi > Mo, or Me < Mi < Mo (the suffixes e, i and o occurring in alphabetical order). In both cases, the median always lies between the arithmetic mean and the mode.

This ordering is linked to the shape of the frequency distribution. In a symmetrical distribution, where the curve is a mirror image on either side of its peak, the mean, median and mode coincide at the same central point.

In a positively skewed distribution, where the curve has a longer tail stretching towards higher values, the mean is pulled above the median, which is itself above the mode, giving the order mean > median > mode.

In a negatively skewed distribution, where the longer tail stretches towards lower values, the order reverses: mean < median < mode.

An approximate empirical relation sometimes used to estimate one average from the other two is Mode = 3Median − 2Mean.

Draw three bell-shaped frequency curves with frequency on the vertical axis and the variable on the horizontal axis.

The first curve is symmetrical, with mean, median and mode marked together at its single peak.

The second curve has a longer tail to the right (positively skewed); mark the mode at the peak, the median just to its right, and the mean furthest right.

The third curve has a longer tail to the left (negatively skewed); mark the mean furthest left, the median in the middle and the mode at the peak on the right.

Which average should be used in a given situation?

The arithmetic mean is the most commonly used average because it is simple to calculate and uses every observation in the data, but for that very reason it is unduly affected by extreme values.

When a distribution contains very large or very small outlying values, the median gives a better summary, because it depends only on the position of the middle items, not on their actual size.

The mode is generally the most useful average for describing qualitative data, such as the most popular shoe size or garment size, where the "typical" value matters more than an arithmetical average.

Both the median and the mode can be found graphically, and both can be computed even for open-ended frequency distributions (where the first or last class has no stated lower or upper limit), which is a situation in which the arithmetic mean cannot be calculated directly.

Returning to the Indian farming example of Balapur village in Buxar district, Bihar, with which this topic began: if a handful of very large landowners among the 50 farmers pulled the mean land holding well above what most farmers actually own, the median (the holding of the middle farmer when all 50 are ranked by size) would describe Baiju's relative position in the village more fairly than the mean, while the mode (the size of holding owned by the largest number of farmers) would show what a typical small farmer in that Indian village holds.

For the same reason, Indian data on farm holdings, family incomes or factory daily wages is often best summarised by more than one average, since the mean, median and mode each answer a slightly different question about the same set of data.

Going beyond the syllabus

These ideas extend the unit and are not needed for the questions on it.

What are quartiles and percentiles?

Quartiles divide an ordered series into four equal parts. The first (lower) quartile Q₁ has 25% of the items below it; the second quartile Q₂ is the median itself, with 50% of items below it; the third (upper) quartile Q₃ has 75% of the items below it and 25% above it.

The region between Q₁ and Q₃ contains the central 50% of the data.

Percentiles divide the distribution into one hundred equal parts, giving ninety-nine dividing positions P₁ to P₉₉, of which P₅₀ is the median.

For an ordered series of N items, the quartile positions are:

Q₁ position = (N + 1)/4 and Q₃ position = 3(N + 1)/4, each giving the item number in the ordered array.

Worked example 11. Ten students scored 22, 26, 14, 30, 18, 11, 35, 41, 12, 32 marks in an examination. Find the lower quartile.

Answer: Arranged in order: 11, 12, 14, 18, 22, 26, 30, 32, 35, 41. Q₁ position = (10 + 1)/4 = 2.75th item = 2nd item + 0.75 × (3rd item − 2nd item) = 12 + 0.75(14 − 12) = 12 + 1.5 = 13.5 marks.

What are the geometric mean and the harmonic mean?

The three averages discussed here, arithmetic mean, median and mode, are the three most commonly used measures of central tendency.

Two further averages, the geometric mean and the harmonic mean, exist and are suitable in certain special situations, although their detailed computation is not part of this topic.

The geometric mean and the harmonic mean are each calculated from the same raw data as the arithmetic mean, but they combine the values by multiplication or by reciprocals respectively, rather than by simple addition, so each gives a different kind of "typical" value from the same set of numbers.

Because every one of the five averages, arithmetic mean, median, mode, geometric mean and harmonic mean, answers the question "what is typical?" in a slightly different way, no single average can be called correct in every situation; the appropriate choice always depends on the purpose of the analysis and on the shape and nature of the data being summarised.

Students who go on to study statistics at a more advanced level will meet the geometric mean and the harmonic mean again, together with methods for deciding which average best suits a particular set of data.

How can the median be located graphically?

Plot the upper class limits on the horizontal axis against the cumulative frequency on the vertical axis to draw a "less than" ogive, and the lower class limits against cumulative frequency for a "more than" ogive.

The point where the two ogives intersect, read down to the horizontal axis, gives the median graphically.

Glossary

• Measure of central value — a single representative value that summarises an entire set of data.
• Arithmetic mean — the sum of all observations divided by the number of observations, denoted X̄.
• Weighted mean — an arithmetic mean in which each value is multiplied by a weight reflecting its relative importance before averaging.
• Median — the value of the middle item of a series arranged in ascending or descending order.
• Mode — the value that occurs most frequently in a series, denoted Mo.
• Assumed mean — a value, existing in the data or not, taken as a trial mean to simplify calculation.
• Step deviation method — a method that divides deviations from the assumed mean by a common factor to reduce the size of figures.
• Discrete series — grouped data in which each distinct value of the variable has its own frequency.
• Continuous series — grouped data presented in class intervals, where the mid-value represents each class.
• Cumulative frequency — the running total of frequencies up to and including a given class.
• Modal class — the class interval in a continuous series with the highest frequency.
• Median class — the class interval in which the N/2th item of a continuous series lies.
• Skewness — the lack of symmetry in a frequency distribution, described as positive or negative according to the direction of the longer tail.

Common errors and misconceptions

• Misconception: The median of a continuous series is located at the (N+1)/2th item, just as for discrete data. Correct: For continuous series, the median class is located at the N/2th item, not the (N+1)/2th item.
• Misconception: The arithmetic mean is always the best average to use. Correct: The mean is unduly affected by extreme values; the median or mode may describe skewed or qualitative data better.
• Misconception: The mode of a series is always unique. Correct: A series may have no mode, one mode (unimodal), two modes (bimodal), or more than two modes (multi-modal).
• Misconception: Any value can be chosen as the assumed mean, but it should be the actual mean for the method to work. Correct: Any value, whether occurring in the data or not, can serve as the assumed mean; a centrally located value merely simplifies the arithmetic.
• Misconception: In the mode formula, D₁ and D₂ can be negative. Correct: D₁ and D₂ are differences in frequency taken by ignoring signs (that is, as positive magnitudes).
• Misconception: Mean, median and mode are always equal in any distribution. Correct: They coincide only in a symmetrical distribution; in skewed distributions they take the order mean > median > mode (positive skew) or mean < median < mode (negative skew).
• Misconception: The median depends on all values in the series, as the mean does. Correct: The median concentrates only on the values of the central items and is not sensitive to every value in the series.

Exam-style questions with model answers

Q1. State the formula for the arithmetic mean of a series of N ungrouped observations. [1 mark]

1. The arithmetic mean is the sum of the values of all observations divided by the number of observations, written X̄ = ΣX/N, where ΣX is the sum of all observations and N is their total number.

Q2. Define the mode and state what is meant by a bimodal series. [2 marks]

1. The mode is the value that is repeated the highest number of times in a series, that is, the most frequently observed data value.
2. A bimodal series is one in which two different values are each repeated the highest number of times, so the data has two modes.

Q3. The marks of five students in an economics test are 40, 50, 55, 78 and 58. Calculate the arithmetic mean by the direct method. [4 marks]

1. Sum of the observations, ΣX = 40 + 50 + 55 + 78 + 58 = 281.
2. Number of observations, N = 5.
3. By the direct method, X̄ = ΣX/N = 281/5 = 56.2.
4. The average mark of the five students in the economics test is therefore 56.2.

Q4. A discrete series gives the values 10, 20, 30, 40 and 50 with frequencies 2, 8, 20, 10 and 5. Calculate the mode and state whether the data is unimodal. [4 marks]

1. For a discrete series, the mode is the value of the variable against which the frequency is highest.
2. The frequencies are 2, 8, 20, 10 and 5, so the maximum frequency is 20.
3. The maximum frequency of 20 belongs to the variable value 30, so the mode is 30.
4. As only one value has the highest frequency, the data is unimodal.

Q5. Twenty persons earn incomes of Rs 10, Rs 20, Rs 30 and Rs 40, with 2, 4, 10 and 4 persons respectively earning each income. Calculate the median income, showing the cumulative frequency column. [4 marks]

1. The cumulative frequencies are 2 for Rs 10, 6 for Rs 20, 16 for Rs 30 and 20 for Rs 40.
2. Total number of persons, N = 20, so the median position is (N + 1)/2 = 21/2 = 10.5th observation.
3. The 10.5th observation falls in the cumulative frequency of 16, which corresponds to an income of Rs 30.
4. Therefore, the median income is Rs 30.

Q6. Daily wages (in Rs) of workers in a factory are: 20 to 25 → 14 workers, 25 to 30 → 28, 30 to 35 → 33, 35 to 40 → 30, 40 to 45 → 20, 45 to 50 → 15, 50 to 55 → 13, 55 to 60 → 7. Calculate the median daily wage. [8 marks]

1. Total number of workers, N = 14 + 28 + 33 + 30 + 20 + 15 + 13 + 7 = 160.
2. Cumulative frequencies are: 20 to 25 → 14, 25 to 30 → 42, 30 to 35 → 75, 35 to 40 → 105, 40 to 45 → 125, 45 to 50 → 140, 50 to 55 → 153, 55 to 60 → 160.
3. The median lies at the N/2th item = 160/2 = 80th item.
4. The 80th item falls in the class 35 to 40, since the cumulative frequency reaches 75 at the end of 30 to 35 and 105 at the end of 35 to 40. So the median class is 35 to 40.
5. For the median class: L (lower limit) = 35, cf (cumulative frequency of the class before it) = 75, f (frequency of the median class) = 30, h (class width) = 5.
6. Median = L + ((N/2 − cf)/f) × h = 35 + ((80 − 75)/30) × 5 = 35 + 0.83 = Rs 35.83.
7. Thus, 50% of the workers earn Rs 35.83 or less, and 50% earn Rs 35.83 or more.

Q7. Distinguish between the median and the mode as measures of central value. [4 marks]

1. The median is the value of the middle item of a series arranged in order, so it depends on the position of items, whereas the mode is the value that occurs most frequently, so it depends on the frequency of repetition.
2. The median always exists and is unique for a given series, whereas a series may have no mode, one mode, or more than one mode.
3. Both the median and the mode are less affected by extreme values than the arithmetic mean.
4. Both can be computed even for open-ended frequency distributions, where the arithmetic mean cannot be calculated directly.

Q8. Explain the relationship between the arithmetic mean, median and mode in a symmetrical distribution, a positively skewed distribution and a negatively skewed distribution. [8 marks]

1. If the arithmetic mean is written Me, the median Mi and the mode Mo, their relative sizes follow one of two orders: Me > Mi > Mo, or Me < Mi < Mo.
2. In both cases the median always lies between the arithmetic mean and the mode.
3. In a symmetrical distribution, where the curve is a mirror image on either side of its peak, the mean, median and mode coincide at the same central point.
4. In a positively skewed distribution, where the curve has a longer tail stretching towards higher values, the mean is pulled above the median, which is itself above the mode, giving the order mean > median > mode.
5. In a negatively skewed distribution, where the longer tail stretches towards lower values, the order reverses to mean < median < mode.
6. The mean is pulled furthest in the direction of the longer tail because it uses every value, including the extreme ones, while the mode stays at the peak of the curve and the median, which depends only on position, lies between them.
7. This relationship is useful for judging the shape of a distribution from its three averages without drawing the full frequency curve.

Key takeaways

• A measure of central value summarises a whole data set with one representative number.
• Arithmetic mean = ΣX/N for ungrouped data, and ΣfX/Σf for grouped data.
• The assumed mean and step deviation methods simplify calculation without changing the final answer.
• A weighted mean assigns different weights to values according to their importance.
• The sum of deviations of all items from the arithmetic mean is always zero.
• The median is the middle value of an ordered series and is not disturbed by extreme values.
• The mode is the most frequently occurring value and suits qualitative or open-ended data.
• Mean, median and mode coincide in a symmetrical distribution but diverge in a predictable order in skewed distributions.

Test yourself

What formula gives the position of the median in an ordered series of N ungrouped items?

Median position = (N + 1)/2, that is, the median is the (N + 1)/2th item of the array arranged in ascending or descending order.

Why is the arithmetic mean described as "unduly affected" by extreme values?

Because it uses the sum of every observation, so one very large or very small value shifts the total, and hence the mean, strongly up or down.

What distinguishes a bimodal distribution from a unimodal one?

A unimodal distribution has a single value that occurs most frequently, while a bimodal distribution has two values that are each the most frequent (joint modes).

In the mode formula for a continuous series, what do D₁ and D₂ represent?

D₁ is the difference between the frequency of the modal class and the frequency of the class preceding it; D₂ is the difference with the class succeeding it, both taken ignoring signs.

Why can the median be calculated for an open-ended frequency distribution but the mean cannot?

The median only needs the cumulative frequency up to the median class, not the actual values at the open ends, whereas the mean needs every value, including those in the open-ended classes.

What is the order of mean, median and mode in a negatively skewed distribution?

Mean is less than median, which is less than mode: the longer tail towards lower values pulls the mean below the median and the mode.

What weight is typically used when calculating a weighted mean for prices of several commodities?

The quantity of each commodity purchased or consumed is typically used as its weight.

How is the mode found in a discrete frequency distribution?

The mode is the value of the variable against which the frequency is highest; for example, if the maximum frequency is 20 and it belongs to the value 30, the mode is 30.