6 Measures of Central Tendency
Measures of central tendency are statistics that describe the center, or the single most typical value, of a dataset. Instead of reading through every observation, a single well-chosen number tells you where the bulk of the data sits. The three classical measures are the mean, the median, and the mode, and each answers a slightly different question about “typical.” This chapter covers all three for raw, ungrouped data, along with two important variants of the mean, the weighted mean and the geometric mean, for situations the simple mean handles poorly. It then extends mean, median, and mode to grouped data organized into the class intervals and frequency tables built in the previous chapter.
6.1 Mean
The arithmetic mean, usually just called the mean or the average, is the sum of all observations divided by the number of observations.
\[\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}\]
- Uses every value: Every observation contributes to the mean, which makes it efficient but also sensitive to outliers.
- Best for: Interval and ratio data with no extreme outliers.
Example
Seven students score the following marks in a class test: 58, 62, 58, 74, 65, 58, 70.
\[\bar{x} = \frac{58+62+58+74+65+58+70}{7} = \frac{445}{7} \approx 63.57\]
6.2 Weighted Mean
The weighted mean is a variant of the arithmetic mean where each value is multiplied by a weight reflecting its relative importance before averaging, rather than treating every observation as equally important. It is the right choice whenever the values being averaged do not all carry the same significance, such as different purchase quantities at different prices, or different credit hours behind different course grades.
\[\bar{x}_w = \frac{\sum (w_i \times x_i)}{\sum w_i}\]
where \(x_i\) is each value and \(w_i\) is the weight attached to it. Note that the grouped mean formula from the next section is, in fact, a special case of the weighted mean: the class marks play the role of \(x_i\) and the class frequencies play the role of \(w_i\).
Example
A retailer buys the same product in three batches during a month: 100 units at ₹12 each, 150 units at ₹15 each, and 250 units at ₹10 each. The simple average of the three prices ignores how many units were bought at each price:
\[\frac{12+15+10}{3} = 12.33\]
The weighted average, using the quantity bought at each price as the weight, gives the true average cost per unit:
\[\bar{x}_w = \frac{(100)(12)+(150)(15)+(250)(10)}{100+150+250} = \frac{5950}{500} = 11.90\]
The weighted mean of ₹11.90 is the figure that actually reflects what the retailer paid per unit, since it accounts for the fact that most units (250 of 500) were bought at the lowest price.
6.3 Geometric Mean
The geometric mean is the \(n\)-th root of the product of \(n\) values. It is the correct average to use whenever the values being averaged are rates of change, such as year-on-year growth rates, interest rates, or inflation rates, situations where the arithmetic mean systematically overstates the true average rate.
\[\text{G.M.} = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n}\]
Never use the arithmetic mean to average percentage growth rates. Because growth compounds multiplicatively, not additively, the arithmetic mean of a series of growth rates is always slightly higher than the true average rate, and the gap widens as the rates get larger or more variable.
Example
A company’s annual revenue growth rate over four years was 5%, 10%, 8%, and 15%, corresponding to growth factors of 1.05, 1.10, 1.08, and 1.15 (a growth factor is \(1 + \frac{\text{rate}}{100}\)).
\[\text{G.M.} = \sqrt[4]{1.05 \times 1.10 \times 1.08 \times 1.15} = \sqrt[4]{1.4345} \approx 1.0944\]
This corresponds to an average growth rate of about 9.44% per year, the rate that, compounded for four years, actually reproduces the company’s real revenue path. The arithmetic mean of the four rates, by contrast, is \(\frac{5+10+8+15}{4} = 9.50\%\), which looks close here but consistently overstates the true average, and the overstatement grows sharply for more volatile growth rates (the classic textbook case: an investment that grows 100% one year and then falls 50% the next is back exactly where it started, a true average growth rate of 0%, yet the arithmetic mean of \(+100\%\) and \(-50\%\) misleadingly suggests \(+25\%\) per year).
6.4 Median
The median is the middle value of a dataset arranged in ascending order. For an odd number of observations it is the single middle value; for an even number, it is the average of the two middle values.
- Not sensitive to outliers: The median depends only on position, not magnitude, making it the preferred measure for skewed data.
- Best for: Ordinal, interval, and ratio data, especially when extreme values are present.
Example
Arranging the same seven marks in ascending order: 58, 58, 58, 62, 65, 70, 74. With 7 (odd) observations, the median is the 4th value: 62.
6.5 Mode
The mode is the value that occurs most frequently in a dataset. A dataset can be unimodal, bimodal, multimodal, or have no mode at all if every value is unique.
- Works for every data type: The mode is the only measure of central tendency valid for nominal data.
- Best for: Categorical data, or finding the single most common value in any dataset.
Example
In the marks 58, 62, 58, 74, 65, 58, 70, the value 58 appears three times, more than any other value, so the mode is 58.
Raw data with many unique values, such as the 30 individual test marks used to build the frequency distribution in the previous chapter, often has no repeated value at all, and therefore no meaningful mode. This is exactly the situation where grouping the data into class intervals, covered next, recovers a usable mode by looking at which class has the highest frequency rather than which single value repeats most.
6.6 Central Tendency for Grouped Data
Once data has been organized into a frequency distribution, as in Chapter 4, each observation’s exact value is no longer available, only the class it falls into and that class’s frequency. All three measures of central tendency have grouped-data formulas that work from the class marks (midpoints) and frequencies alone. The worked examples below reuse the exact frequency table built in Chapter 4: the marks of 30 students in a statistics test, grouped into seven classes of width 10.
| Class Interval | Class Mark (\(x\)) | Frequency (\(f\)) | Cumulative Frequency |
|---|---|---|---|
| 30–40 | 35 | 2 | 2 |
| 40–50 | 45 | 4 | 6 |
| 50–60 | 55 | 6 | 12 |
| 60–70 | 65 | 6 | 18 |
| 70–80 | 75 | 5 | 23 |
| 80–90 | 85 | 4 | 27 |
| 90–100 | 95 | 3 | 30 |
6.6.1 Mean of Grouped Data
Direct Method: Multiply each class mark by its frequency, sum these products, and divide by the total frequency \(N\).
\[\bar{x} = \frac{\sum f_i x_i}{N}\]
Step-Deviation Method: A quicker shortcut for hand calculation. Pick an assumed mean \(A\) (usually the class mark of the middle class), compute \(d_i' = \frac{x_i - A}{h}\) for each class where \(h\) is the class width, then:
\[\bar{x} = A + \left(\frac{\sum f_i d_i'}{N}\right) \times h\]
Example
Using the direct method with \(A = 65\) and \(h = 10\) for the step-deviation check:
\[\bar{x} = \frac{(2)(35)+(4)(45)+(6)(55)+(6)(65)+(5)(75)+(4)(85)+(3)(95)}{30} = \frac{1970}{30} \approx 65.67\]
Both methods agree: the grouped mean is 65.67, close to but not identical to the ungrouped mean of the same 30 raw marks (65.2), the small difference coming from replacing each observation with its class midpoint.
6.6.2 Median of Grouped Data
\[\text{Median} = L + \left(\frac{\frac{N}{2} - cf}{f}\right) \times h\]
where \(L\) is the lower boundary of the median class (the class containing the \(\frac{N}{2}\)-th observation), \(cf\) is the cumulative frequency before the median class, \(f\) is the frequency of the median class, and \(h\) is the class width.
Example
\(N = 30\), so \(\frac{N}{2} = 15\). From the cumulative frequency column, the 60–70 class is the first to reach a cumulative frequency of 15 or more (its cumulative frequency is 18), so it is the median class, with \(L = 60\), \(cf = 12\) (the cumulative frequency of the class before it), \(f = 6\), and \(h = 10\).
\[\text{Median} = 60 + \left(\frac{15-12}{6}\right) \times 10 = 60 + 5 = 65\]
6.6.3 Mode of Grouped Data
\[\text{Mode} = L + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h\]
where \(L\) is the lower boundary of the modal class (the class with the highest frequency), \(f_1\) is the frequency of the modal class, \(f_0\) is the frequency of the class before it, \(f_2\) is the frequency of the class after it, and \(h\) is the class width.
Example
The highest frequency is 6, shared by the 50–60 and 60–70 classes; by convention the first such class is taken as the modal class, so \(L = 50\), \(f_1 = 6\), \(f_0 = 4\), \(f_2 = 6\), and \(h = 10\).
\[\text{Mode} = 50 + \left(\frac{6-4}{2(6)-4-6}\right) \times 10 = 50 + \left(\frac{2}{2}\right) \times 10 = 60\]
6.7 When to Use Each Measure
- Mean: Best for symmetric data without outliers, since it uses every value and has strong mathematical properties for further analysis.
- Weighted Mean: Best when observations don’t carry equal importance, such as averaging prices across unequal purchase quantities or grades across unequal credit hours.
- Geometric Mean: Best for averaging rates of change over time, such as growth rates, interest rates, or inflation, where the arithmetic mean would overstate the true average.
- Median: Best for skewed data or data with outliers, such as income or property prices, since a few extreme values cannot pull it away from the bulk of the data.
- Mode: Best for categorical data, or when the business question is specifically “what is the most common value,” such as the best-selling product size.
A common mistake is reporting only the mean without checking whether the data is skewed. If mean and median differ substantially, the data is skewed, and the median (or both figures together) tells a more honest story than the mean alone. Averaging a small sample with even one extreme outlier is the classic way a mean misleads.
Looking Ahead
Knowing where the center of a dataset lies is only half the picture. Two datasets can share the exact same mean and still behave very differently, one tightly clustered, one wildly spread out. The next chapter covers measures of dispersion, which quantify exactly that spread, again for both raw and grouped data.
Summary
| Concept | Description |
|---|---|
| Core Idea | |
| Central Tendency | A single representative value that summarises where the bulk of a dataset sits |
| Mean | The arithmetic average, the sum of all observations divided by their count |
| Other Types of Mean | |
| Weighted Mean | A mean where each value is multiplied by a weight of relative importance before averaging, used when observations are not equally significant |
| Geometric Mean | The n-th root of the product of n values, the correct average for rates of change such as growth or interest rates |
| Median and Mode | |
| Median | The middle value of an ordered dataset, or the average of the two middle values when the count is even |
| Mode | The value that occurs most frequently in a dataset |
| Robustness to Outliers | |
| Mean Sensitivity to Outliers | The mean is pulled toward extreme values, since every observation contributes to it |
| Median and Outliers | The median depends only on position, so it is unaffected by how extreme the outlying values are |
| Mode and Data Types | The mode is the only central tendency measure that works for nominal, non-numeric data |
| Grouped Mean | |
| Class Mark | The midpoint of a class interval, used to represent every observation within that class |
| Grouped Mean (Direct Method) | Sum of each class mark times its frequency, divided by the total frequency |
| Grouped Mean (Step-Deviation Method) | A shortcut using an assumed mean and standardised class deviations, mathematically identical to the direct method |
| Grouped Median | |
| Median Class | The class interval containing the N over 2 th observation, identified from the cumulative frequency column |
| Grouped Median Formula | L plus the ratio of the shortfall to N over 2 over the median class frequency, all times the class width |
| Grouped Mode | |
| Modal Class | The class interval with the highest frequency |
| Grouped Mode Formula | L plus a weighted ratio of the modal class frequency against its neighbours, all times the class width |
| Choosing the Right Measure | |
| Symmetrical Data | The mean is the preferred measure when the distribution has no extreme outliers |
| Skewed Data | The median is the preferred measure when a few extreme values would distort the mean |
| Categorical Data | The mode is the preferred measure when the variable is categorical or attribute-based |