What this chapter is about
Statistics is the branch of mathematics that deals with collecting, organising, analysing and interpreting numerical data. In Class 10, you move beyond simple averages to learn how to find the mean, median and mode for data that is grouped into class intervals. This is essential because real-world data — exam scores of hundreds of students, wages of workers, heights of people — is usually presented in grouped form.
You will learn three methods to calculate the mean of grouped data, understand how to locate the median class and apply a formula to find the exact median, and identify the modal class to determine the mode. These measures of central tendency help summarise large datasets with a single representative value.
After studying this chapter, you should be able to construct a cumulative frequency table, draw cumulative frequency curves (ogives), and use graphical methods to estimate the median. You will also understand when each measure — mean, median or mode — is most appropriate to describe data.
Key ideas
- Grouped data is data organised into class intervals (e.g., 10–20, 20–30) along with their frequencies; the exact values within each class are not known.
- The class mark (or mid-value) of a class interval is calculated as (lower limit + upper limit) / 2 and represents all observations in that class.
- The mean of grouped data can be found using three methods: Direct Method, Assumed Mean Method and Step Deviation Method — all give the same answer but differ in computational ease.
- The median is the middle value of ordered data; for grouped data, we first find the median class (the class containing the n/2th observation) and then apply a formula.
- The mode is the most frequently occurring value; for grouped data, the modal class is the class with the highest frequency.
- Cumulative frequency is the running total of frequencies up to each class; it helps locate the median class and draw ogives.
- An ogive (cumulative frequency curve) can be of two types: less than type and more than type; their intersection point gives the median.
Formulas and facts to remember
Mean by Direct Method: Mean = Σ(fᵢxᵢ) / Σfᵢ where fᵢ is frequency and xᵢ is class mark.
Mean by Assumed Mean Method: Mean = a + (Σfᵢdᵢ / Σfᵢ) where a is assumed mean and dᵢ = xᵢ − a.
Mean by Step Deviation Method: Mean = a + h × (Σfᵢuᵢ / Σfᵢ) where h is class size and uᵢ = (xᵢ − a) / h.
Median of Grouped Data: Median = l + [(n/2 − cf) / f] × h where l = lower limit of median class, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class size.
Mode of Grouped Data: Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h 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.
Empirical relationship: 3 × Median = Mode + 2 × Mean (approximate relationship).
Worked examples
Example 1: Finding mean using Step Deviation Method
The weekly pocket money (in rupees) of 50 students is given below. Find the mean.
- Pocket Money: Students · 50–100: 8 · 100–150: 12 · 150–200: 18 · 200–250: 7 · 250–300: 5
Solution: Take assumed mean a = 175 (class mark of middle class) and h = 50.
- Class: 50–100 · fᵢ: 8 · xᵢ: 75 · uᵢ = (xᵢ − 175)/50: −2 · fᵢuᵢ: −16
- Class: 100–150 · fᵢ: 12 · xᵢ: 125 · uᵢ = (xᵢ − 175)/50: −1 · fᵢuᵢ: −12
- Class: 150–200 · fᵢ: 18 · xᵢ: 175 · uᵢ = (xᵢ − 175)/50: 0 · fᵢuᵢ: 0
- Class: 200–250 · fᵢ: 7 · xᵢ: 225 · uᵢ = (xᵢ − 175)/50: 1 · fᵢuᵢ: 7
- Class: 250–300 · fᵢ: 5 · xᵢ: 275 · uᵢ = (xᵢ − 175)/50: 2 · fᵢuᵢ: 10
- Class: Total · fᵢ: 50 · xᵢ: · uᵢ = (xᵢ − 175)/50: · fᵢuᵢ: −11
Mean = 175 + 50 × (−11/50) = 175 − 11 = 164 rupees.
Example 2: Finding median of grouped data
The marks obtained by 60 students are given below. Find the median.
- Marks: Students · 0–10: 5 · 10–20: 10 · 20–30: 20 · 30–40: 15 · 40–50: 10
Solution: n = 60, so n/2 = 30.
Cumulative frequencies: 5, 15, 35, 50, 60.
The median class is 20–30 (cumulative frequency first exceeds 30).
Here, l = 20, cf = 15, f = 20, h = 10.
Median = 20 + [(30 − 15) / 20] × 10 = 20 + (15/20) × 10 = 20 + 7.5 = 27.5 marks.
Example 3: Finding mode of grouped data
The ages of 100 patients visiting a clinic are recorded below. Find the mode.
- Age (years): Patients · 10–20: 12 · 20–30: 25 · 30–40: 38 · 40–50: 18 · 50–60: 7
Solution: Modal class is 30–40 (highest frequency = 38).
Here, l = 30, f₁ = 38, f₀ = 25, f₂ = 18, h = 10.
Mode = 30 + [(38 − 25) / (2 × 38 − 25 − 18)] × 10 Mode = 30 + [13 / (76 − 43)] × 10 Mode = 30 + (13/33) × 10 = 30 + 3.94 ≈ 33.94 years.
Common mistakes
- Using class limits instead of class marks in mean calculation → Always find class mark as (lower + upper)/2 before multiplying by frequency.
- Confusing cumulative frequency with frequency when finding median → cf is the total before the median class, not the frequency of median class.
- Taking the wrong class as modal class → Modal class is the class with highest frequency, not the highest class mark.
- Forgetting to subtract cf (cumulative frequency before median class) in the median formula → The formula requires (n/2 − cf), not just n/2.
- Using unequal class intervals without adjustment → Ensure all classes have equal width, or adjust frequencies accordingly.
Quick revision
- Mean uses class marks; three methods (Direct, Assumed Mean, Step Deviation) all give the same answer.
- Median class contains the (n/2)th observation; use cumulative frequency to locate it.
- Mode formula needs the modal class (highest frequency) and frequencies of its neighbours.
- Ogives are cumulative frequency curves; the x-coordinate where less-than and more-than ogives meet gives the median.
- Always check: Σfᵢ = n (total observations) before applying any formula.