Big picture
A frequency distribution still contains many numbers. Often we want one value that represents the whole group, such as the average income of a village or the typical size of a family. This value is a measure of central tendency (an average), because values tend to cluster around it.
The three main averages are the arithmetic mean, the median and the mode. Each is calculated differently for individual, discrete and continuous series, and each suits different kinds of data. Numericals carry heavy marks, so practise each formula.
Learning goals
Calculate the arithmetic mean by the direct, assumed mean and step deviation methods, and the weighted mean.
Calculate the median and quartiles for all three types of series.
Calculate the mode for all three types of series.
Explain the merits and limitations of each average.
Use the empirical relationship between mean, median and mode.
Choose the most suitable average for a given situation.
Core ideas
1. Arithmetic mean
The arithmetic mean is the sum of all values divided by the number of values. It is written as X̄.
Series | Direct method | Assumed mean method | Step deviation method |
|---|---|---|---|
Individual | X̄ = ΣX ÷ N | X̄ = A + Σd ÷ N, where d = X − A | |
Discrete | X̄ = ΣfX ÷ Σf | X̄ = A + Σfd ÷ Σf | X̄ = A + (Σfd′ ÷ Σf) × c, where d′ = (X − A) ÷ c |
Continuous | Use mid-points (m): X̄ = Σfm ÷ Σf | X̄ = A + Σfd ÷ Σf, d = m − A | X̄ = A + (Σfd′ ÷ Σf) × c |
Individual series. Marks of five students: 40, 50, 55, 78, 58. X̄ = 281 ÷ 5 = 56.2.
Discrete series.
Marks (X) | Students (f) | fX |
|---|---|---|
10 | 3 | 30 |
20 | 5 | 100 |
30 | 8 | 240 |
40 | 4 | 160 |
Total | 20 | 530 |
X̄ = 530 ÷ 20 = 26.5.
Continuous series (direct and step deviation).
Class | f | Mid-point m | fm | d′ = (m − 25) ÷ 10 | fd′ |
|---|---|---|---|---|---|
0–10 | 5 | 5 | 25 | −2 | −10 |
10–20 | 8 | 15 | 120 | −1 | −8 |
20–30 | 12 | 25 | 300 | 0 | 0 |
30–40 | 10 | 35 | 350 | 1 | 10 |
40–50 | 5 | 45 | 225 | 2 | 10 |
Total | 40 | 1,020 | 2 |
Direct: X̄ = 1,020 ÷ 40 = 25.5. Step deviation: X̄ = 25 + (2 ÷ 40) × 10 = 25.5. Both methods give the same answer; the step deviation method simply makes the arithmetic easier.
Properties: the sum of deviations of values from their mean is always zero: Σ(X − X̄) = 0. The mean is affected by every value, including extreme ones.
Weighted arithmetic mean. When items differ in importance, each is given a weight (w): X̄w = ΣwX ÷ Σw.
Example. Prices per kg: wheat ₹30 (weight 4), rice ₹40 (weight 3), pulses ₹100 (weight 1). X̄w = (120 + 120 + 100) ÷ 8 = 340 ÷ 8 = ₹42.50. The simple mean of the three prices, ₹56.67, would overstate the cost, since pulses are bought in small quantities.
Merits of the mean | Limitations |
|---|---|
Simple to understand and calculate | Affected by extreme values (a few very rich people raise average income) |
Based on all observations | Can't be found for open-ended classes without assumptions |
Rigidly defined; capable of further algebraic treatment | Can't be found graphically |
Stable from sample to sample | May give a value not in the data (2.4 children) |
2. Median
The median is the middle value when data are arranged in ascending or descending order. It divides the series into two equal parts. It is a positional average.
Individual series. Arrange the values in order. Median = size of the ((N + 1) ÷ 2)th item.
Marks 40, 50, 55, 58, 78 (N = 5): the 3rd item, 55.
Values 10, 15, 20, 25 (N = 4): the 2.5th item, the average of the 2nd and 3rd = (15 + 20) ÷ 2 = 17.5.
Discrete series. Find cumulative frequencies; median = size of the ((N + 1) ÷ 2)th item. In the discrete example above (N = 20), cumulative frequencies are 3, 8, 16, 20; the 10.5th item lies in the group with cumulative frequency 16, so the median is 30.
Continuous series.
Median = L + ((N ÷ 2 − cf) ÷ f) × h
where L is the lower limit of the median class, cf the cumulative frequency of the class before it, f the frequency of the median class, and h the class width.
Using the continuous series above (N = 40): cumulative frequencies are 5, 13, 25, 35, 40. N ÷ 2 = 20, which falls in the class 20–30 (cf rises from 13 to 25). L = 20, cf = 13, f = 12, h = 10.
Median = 20 + ((20 − 13) ÷ 12) × 10 = 20 + 5.83 = 25.83.
Quartiles divide a series into four equal parts. Q1 (lower quartile) has 25% of values below it; Q3 (upper quartile) has 75% below it; Q2 is the median.
Q1 = L + ((N ÷ 4 − cf) ÷ f) × h and Q3 = L + ((3N ÷ 4 − cf) ÷ f) × h
Same series: Q1 item = 40 ÷ 4 = 10th, in class 10–20: Q1 = 10 + ((10 − 5) ÷ 8) × 10 = 16.25. Q3 item = 30th, in class 30–40: Q3 = 30 + ((30 − 25) ÷ 10) × 10 = 35.
Merits of the median | Limitations |
|---|---|
Not affected by extreme values | Requires arranging data in order |
Can be found for open-ended classes | Not based on all observations |
Can be found graphically (ogives) | Not capable of further algebraic treatment |
Suitable for qualitative data that can be ranked (intelligence, honesty) | Affected more by sampling fluctuations than the mean |
3. Mode
The mode is the value that occurs most often, the most typical value.
Individual series: the value repeated most often. In 4, 6, 6, 7, 8, 6, 9 the mode is 6.
Discrete series: the value with the highest frequency. In the discrete example above, 30 has the highest frequency (8), so the mode is 30. (When frequencies are close or irregular, a grouping table is used.)
Continuous series.
Mode = L + ((f1 − f0) ÷ (2f1 − f0 − f2)) × h
where L is the lower limit of the modal class (the class with the highest frequency), f1 its frequency, f0 the frequency of the class before it, f2 the frequency of the class after it, and h the class width.
Using the continuous series above: the modal class is 20–30 (f1 = 12, f0 = 8, f2 = 10, L = 20, h = 10).
Mode = 20 + ((12 − 8) ÷ (24 − 8 − 10)) × 10 = 20 + (4 ÷ 6) × 10 = 26.67.
The mode can also be found graphically from a histogram, by drawing lines from the corners of the tallest bar to the adjacent bars.
Merits of the mode | Limitations |
|---|---|
Easy to understand; often the most typical value | Not rigidly defined; a series may have two or more modes, or none |
Not affected by extreme values | Not based on all observations |
Can be found graphically | Not capable of further algebraic treatment |
Used for qualitative data (most popular shoe size or colour) |
4. Relationship between mean, median and mode
For a moderately asymmetrical distribution, the empirical relationship is:
Mode = 3 Median − 2 Mean
Check with the series above: 3 × 25.83 − 2 × 25.5 = 77.49 − 51 = 26.49, close to the calculated mode of 26.67.
Example. If mean = 40 and median = 38, then mode = 3 × 38 − 2 × 40 = 114 − 80 = 34.
In a symmetrical distribution, mean = median = mode.
5. Choosing the right average
Situation | Suitable average | Reason |
|---|---|---|
Average marks of a class, average output | Mean | Uses all values; good for further calculations |
Average income where a few people are very rich | Median | Not affected by extreme values |
Most popular size of shoes or readymade shirts | Mode | Shows the most common value |
Open-ended classes ("above 50") | Median or mode | The mean needs every class closed |
Qualitative data that can be ranked | Median | Positional |
A reliable problem-solving routine
Identify the series: individual, discrete or continuous.
For the mean, use mid-points for continuous series; use step deviation when values are large and class widths equal.
For the median and quartiles, find cumulative frequencies, locate the class, and apply the formula with the correct cf (of the class before).
For the mode, locate the class with the highest frequency, and use f0 and f2 from its neighbours.
Check reasonableness: in most distributions, the mean, median and mode are close.
Connections that matter
Organisation and presentation of data produces the frequency distributions used here; ogives give the median graphically.
Correlation uses the mean in Karl Pearson's coefficient.
Index numbers use the weighted mean.
Common mistakes
Using class limits instead of mid-points for the mean of a continuous series.
Using the cf of the median class instead of the class before it.
Forgetting to arrange data in order before finding the median of an individual series.
Choosing the mean for income data with extreme values. The median is better.
Reversing the empirical formula. Mode = 3 Median − 2 Mean.
Revision checklist
I can calculate the mean by all three methods and the weighted mean.
I can calculate the median and quartiles for every type of series.
I can calculate the mode for every type of series.
I can state the merits and limitations of each average.
I can apply Mode = 3 Median − 2 Mean.
I can choose the right average for a situation.