Chapter Introduction
A measure of central tendency is a single value that represents the centre or typical value of a dataset. It tells you where the "middle" of the data lies. The three most important measures are the arithmetic mean, the median, and the mode. Each has its own formula, advantages, and situations where it is most appropriate. This chapter is one of the most calculation-heavy chapters in the syllabus and carries significant weight in the CBSE exam. Mastering the formulas for individual, discrete, and continuous series is essential.
Every exam paper will have at least one numerical question from this chapter. Practising all three measures across all three series types is non-negotiable.
Key Definitions
| Term | Definition |
|---|---|
| Central tendency | The tendency of data to cluster around a central value; the single value representing the entire dataset |
| Arithmetic mean | The sum of all values divided by the number of observations; the most common average |
| Median | The middle value when data is arranged in ascending or descending order; divides data into two equal halves |
| Mode | The value that occurs most frequently in the dataset |
| Weighted mean | A mean where each value is multiplied by its weight (importance) before averaging |
| Combined mean | The mean of two or more groups combined, calculated from their individual means and sizes |
| Quartiles | Values that divide the data into four equal parts (Q1, Q2=median, Q3) |
| Percentiles | Values that divide the data into 100 equal parts |
| Cumulative frequency (cf) | Running total of frequencies used to locate median and quartile classes |
| Median class | The class interval where cumulative frequency first exceeds N/2 |
| Modal class | The class interval with the highest frequency |
Key Formulas
Arithmetic Mean
| Type | Series | Formula |
|---|---|---|
| Simple mean | Individual | x-bar = Sum of x / n |
| Simple mean | Discrete | x-bar = Sum of fx / Sum of f |
| Simple mean (direct) | Continuous | x-bar = Sum of fx / Sum of f (where x = class mark) |
| Shortcut (assumed mean) | Any | x-bar = A + (Sum of fd / Sum of f), where d = x - A |
| Step-deviation | Continuous | x-bar = A + (Sum of fd' / Sum of f) x h, where d' = (x - A) / h |
| Weighted mean | Weighted data | x-bar(w) = Sum of wx / Sum of w |
| Combined mean | Two groups | x-bar(combined) = (n1 x x-bar1 + n2 x x-bar2) / (n1 + n2) |
| Corrected mean | After error correction | x-bar(corrected) = (Sum of x - wrong value + correct value) / n |
Median
| Series | Formula |
|---|---|
| Individual (odd n) | Md = value at (n+1)/2 th position |
| Individual (even n) | Md = average of values at n/2 and (n/2 + 1)th positions |
| Discrete | Md = value where cumulative frequency first reaches or exceeds N/2 |
| Continuous | Md = L + [(N/2 - cf) / f] x h |
Where: L = lower limit of median class, cf = cumulative frequency before median class, f = frequency of median class, h = class width
Mode
| Series | Formula |
|---|---|
| Individual | Mo = value with highest frequency |
| Discrete | Mo = value with highest frequency (or group with highest frequency) |
| Continuous | Mo = L + [(f1 - f0) / (2f1 - f0 - f2)] x h |
| Empirical relation | Mo = 3Md - 2 x x-bar |
Where: L = lower limit of modal class, f1 = frequency of modal class, f0 = frequency of class before modal class, f2 = frequency of class after modal class, h = class width
Quartiles (Continuous Series)
| Quartile | Formula |
|---|---|
| Q1 | Q1 = L + [(N/4 - cf) / f] x h (class where cf first exceeds N/4) |
| Q3 | Q3 = L + [(3N/4 - cf) / f] x h (class where cf first exceeds 3N/4) |
Percentiles (Continuous Series)
Pk = L + [(kN/100 - cf) / f] x h (class where cf first exceeds kN/100)
Detailed Concepts
Arithmetic Mean — All Methods Explained
Direct method: Best when values are small and few.
x-bar = Sum of x / n (individual series) x-bar = Sum of fx / N (discrete/continuous, where x = class mark)
Shortcut (assumed mean) method: Best when values are large. Take A as a convenient value near the centre. d = x - A. Then:
x-bar = A + (Sum of fd / N)
The result is the same as the direct method. The advantage is that d values are smaller, reducing calculation errors.
Step-deviation method: Best when values are large and class width is constant. d' = (x - A) / h. Then:
x-bar = A + (Sum of fd' / N) x h
This simplifies calculations greatly for continuous series with equal class width because d' values are typically small integers like -3, -2, -1, 0, 1, 2, 3.
Combined mean: When two groups with sizes n1, n2 and means x-bar1, x-bar2 are combined:
x-bar(combined) = (n1 x x-bar1 + n2 x x-bar2) / (n1 + n2)
This extends to three or more groups: just add more n x x-bar terms in the numerator and more n terms in the denominator.
Corrected mean: If a wrong value was used in calculating the mean:
- Corrected sum = old sum - wrong value + correct value
- Corrected mean = corrected sum / n
Weighted mean: When items have different importance:
x-bar(w) = Sum of wx / Sum of w
Example: A student scores 80 in Maths (weight 4) and 60 in English (weight 2). Weighted mean = (4x80 + 2x60) / (4+2) = (320+120)/6 = 73.3
Mean vs Median vs Mode Comparison
| Feature | Mean | Median | Mode |
|---|---|---|---|
| Definition | Sum of values / number of values | Middle value when data is ordered | Most frequently occurring value |
| Calculation | Uses all values | Uses position only | Uses frequency only |
| Affected by extreme values | Yes — highly sensitive | No — resistant to outliers | No — not affected |
| Rigid formula | Yes | Yes | Yes (continuous); may not be unique |
| Applicable to qualitative data | No | Yes (ordinal data) | Yes (nominal data) |
| Mathematical treatment | Possible (used in further analysis) | Not possible | Not possible |
| Graphical determination | Not possible | From ogive (intersection) | From histogram (rectangle method) |
| Existence | Always exists and is unique | Always exists and is unique | May not exist or may not be unique |
| Best for | Normal distribution, symmetric data | Skewed data, data with outliers | Data with a clear peak frequency |
| Sum of deviations | Sum of (x - x-bar) = 0 (always) | Minimises sum of absolute deviations | Minimises maximum frequency |
Graphical Location of Median and Mode
Median from ogive:
1. Draw the "less than" ogive (cumulative frequency curve) 2. Locate N/2 on the Y-axis 3. Draw a horizontal line from N/2 to the curve 4. Drop a vertical from the intersection to the X-axis 5. The X-axis value is the median
Alternatively, draw both ogives (less than and more than) on the same graph. Their intersection, projected to the X-axis, gives the median.
Mode from histogram:
1. Draw the histogram 2. Identify the tallest rectangle (modal class) 3. Join the top-left corner of the modal rectangle to the top-left corner of the next rectangle 4. Join the top-right corner of the modal rectangle to the top-right corner of the previous rectangle 5. The intersection of these diagonal lines, projected to the X-axis, gives the mode
Quartiles and Percentiles
Quartiles divide data into four equal parts:
- Q1 (First quartile) — 25% data below, 75% above. Also called the lower quartile.
- Q2 (Second quartile) — Same as the median. 50% above, 50% below.
- Q3 (Third quartile) — 75% data below, 25% above. Also called the upper quartile.
In continuous series: Q1 class is where cumulative frequency first exceeds N/4. Q3 class is where cumulative frequency first exceeds 3N/4. The formulas are analogous to the median formula.
Percentiles work the same way: Pk class is where cumulative frequency first exceeds kN/100. For example, P10 = class where cf first exceeds 10% of N. P90 = class where cf first exceeds 90% of N.
Worked Example: Step-Deviation Method for Mean
| Class | f | x (class mark) | d'=(x-25)/5 | fd' | fd' squared |
|---|---|---|---|---|---|
| 0-10 | 4 | 5 | -4 | -16 | 64 |
| 10-20 | 8 | 15 | -2 | -16 | 32 |
| 20-30 | 12 | 25 | 0 | 0 | 0 |
| 30-40 | 6 | 35 | 2 | 12 | 24 |
| 40-50 | 5 | 45 | 4 | 20 | 80 |
| Total | 35 | 0 | 200 |
A = 25, h = 5, N = 35
x-bar = A + (Sum of fd' / N) x h = 25 + (0/35) x 5 = 25 + 0 = 25
Worked Example: Finding Median in Continuous Series
Using the same data: N = 35, so N/2 = 17.5
| Class | f | Cumulative frequency |
|---|---|---|
| 0-10 | 4 | 4 |
| 10-20 | 8 | 12 |
| 20-30 | 12 | 24 |
| 30-40 | 6 | 30 |
| 40-50 | 5 | 35 |
Median class: cf first exceeds 17.5 → class 20-30 (cf = 24 exceeds 17.5, previous cf = 12)
Md = L + [(N/2 - cf) / f] x h = 20 + [(17.5 - 12) / 12] x 10 = 20 + (5.5/12) x 10 = 20 + 4.58 = 24.58
Important Points for Exams
1. Arithmetic mean = Sum of x / n (individual) or Sum of fx / N (frequency series). 2. The shortcut and step-deviation methods give the same result as the direct method. 3. Combined mean = (n1 x x-bar1 + n2 x x-bar2) / (n1 + n2). 4. Median for continuous series: Md = L + [(N/2 - cf) / f] x h. Identify the median class first. 5. Mode for continuous series: Mo = L + [(f1 - f0) / (2f1 - f0 - f2)] x h. Identify the modal class first. 6. Empirical relation: Mode = 3 Median - 2 Mean. Use only for moderately skewed distributions. 7. The median is the best average when data has extreme values because it is not affected by them. 8. The mean is used when further statistical analysis is needed. 9. The mode is the only average applicable to qualitative (nominal) data. 10. Quartiles: Q1 at N/4, Q2 (median) at N/2, Q3 at 3N/4. 11. The sum of deviations from the mean is always zero: Sum of (x - x-bar) = 0.
Common Mistakes
| Mistake | Why it is wrong | Correct approach |
|---|---|---|
| Using N/2 instead of (N+1)/2 for median in individual series (odd n) | For individual data with odd n, the median position is (n+1)/2, not n/2 | Md = value at (n+1)/2th position |
| Forgetting to calculate cumulative frequency before finding median class | The median class is found using cumulative frequency, not simple frequency | Always compute cf column first |
| Using the modal class as the mode | The modal class is the class with highest frequency, but the mode is a specific value within it | Apply the formula Mo = L + [(f1-f0)/(2f1-f0-f2)] x h |
| Confusing N with n in formulas | N = total frequency (Sum of f); n = number of observations in individual series | In frequency distributions, always use N = Sum of f |
| Using the empirical relation when the distribution is symmetric | When symmetric, mean = median = mode, so the formula gives the correct but unnecessary result | Use empirical relation only when distribution is moderately skewed |
| Not adjusting for unequal class width in histogram-based mode | The formula assumes equal class widths | If widths are unequal, adjust frequencies before identifying modal class |
| Forgetting to multiply by h in the step-deviation method | x-bar = A + (Sum of fd' / N) x h — the h is easy to forget | Always multiply Sum of fd' / N by h before adding A |
NCERT Exercise Focus
Focus on these NCERT exercise questions while preparing:
1. "Calculate arithmetic mean using direct, shortcut, and step-deviation methods" — Practice all three for the same data. 2. "Find median from the given continuous series" — Practice identifying the median class and applying the formula. 3. "Find mode from the given continuous series" — Practice identifying the modal class and applying the formula. 4. "Given mean = 50, median = 48, find mode using empirical relation" — Mo = 3(48) - 2(50) = 144 - 100 = 44. 5. "Calculate combined mean of two groups" — Practice the combined mean formula.
Exam Tips
1. Numerical problems are the most tested — Practice computing mean (all three methods), median, and mode for continuous series. This is the highest-weighted topic. 2. Always show the formula first, then substitute values, then compute. Examiners award step marks. 3. Step-deviation method — The most efficient method for long continuous series. Master it. 4. Combined mean — A quick 2-3 mark question. The formula is straightforward. 5. Median class vs Modal class — Median class is where cf exceeds N/2; modal class has highest frequency. Do not confuse them. 6. Empirical relation — Memorise Mode = 3Md - 2Mean. You may be asked to find mode given mean and median, or vice versa. 7. Graphical methods — Be prepared to describe how to locate median from ogive and mode from histogram. These are 3-mark questions. 8. Corrected mean — A common 2-mark numerical. Corrected sum = old sum - wrong + correct, then divide by n.
Practice Questions for Self-Test
1. Calculate mean, median, and mode for a given continuous frequency distribution. (5 marks) 2. Distinguish between arithmetic mean, median, and mode. (3 marks) 3. What is the empirical relationship between mean, median, and mode? (2 marks) 4. Calculate combined mean of two groups with given means and sizes. (2 marks) 5. Explain the step-deviation method for calculating arithmetic mean with an example. (5 marks) 6. What is weighted mean? How is it different from simple mean? (3 marks) 7. How is the median located graphically from an ogive? (3 marks) 8. What are quartiles? How are Q1 and Q3 calculated in a continuous series? (3 marks) 9. Which measure of central tendency is best for data with extreme values? Why? (2 marks) 10. A wrong value of 50 was used instead of 40. The original mean was 30 for 20 observations. Find the corrected mean. (2 marks)
Quick Revision Notes
- Mean (individual): x-bar = Sum of x / n. Mean (frequency): x-bar = Sum of fx / N.
- Shortcut method: x-bar = A + Sum of fd / N. Step-deviation: x-bar = A + (Sum of fd' / N) x h.
- Combined mean: (n1 x x-bar1 + n2 x x-bar2) / (n1 + n2).
- Corrected mean: (old sum - wrong + correct) / n.
- Weighted mean: Sum of wx / Sum of w.
- Median (continuous): Md = L + [(N/2 - cf) / f] x h. First find median class (cf exceeds N/2).
- Mode (continuous): Mo = L + [(f1 - f0) / (2f1 - f0 - f2)] x h. First find modal class (highest f).
- Empirical relation: Mo = 3Md - 2Mean.
- Quartiles: Q1 at N/4, Q2 = Median at N/2, Q3 at 3N/4.
- Median: best with outliers. Mean: best for further analysis. Mode: best for qualitative data.
- Graphical median: from ogive intersection. Graphical mode: from histogram diagonals.
- Sum of deviations from mean = 0 (always remember this property).