Class 10 Maths — Chapter 13: Statistics

By RAAM SETU | NEET, SSC, Railway & Police Bharti Focused Notes 📘

📘 Introduction

This chapter covers Statistics including Mean, Median, Mode, Range, and Graphical Representation. Learn formulas, solved examples, and practice questions. Essential for NEET, SSC, Railway, Police Bharti and Class 10 exams.

🔹 Key Formulas

📊 Chapter Summary Table

Concept Formula / Definition Use
MeanΣx / nAverage value
MedianMiddle valueCentral tendency
ModeMost frequent valueMost representative value
RangeMax − MinSpread of data

🧮 Practice Questions

  1. Find the mean of the data: 5, 7, 12, 15, 18.
  2. Find the median of the data: 3, 8, 9, 12, 15.
  3. Find the mode of the data: 4, 4, 6, 7, 7, 7, 9.
  4. Calculate the range of the data: 12, 7, 9, 20, 15.
  5. Prepare a frequency table and identify mean, median, and mode.

Statistics — Class 10 (Advanced Notes)

Data handling • Frequency tables • Mean, Median, Mode • Histogram & Ogive • Measures of dispersion • Practice
Jump to Practice

1. Types of Data & Organisation

Data may be qualitative (categories) or quantitative (numbers). Quantitative data can be discrete (countable) or continuous (measurable).

Organise raw data into frequency distribution tables for easier analysis: list classes (for continuous data) and their frequencies.

2. Frequency Distribution & Class Intervals

Choose class width (h) and starting point to cover the data range. For class interval notation prefer closed-open style: [a, b) so no overlap.

Cumulative frequency (cf) = running total of frequencies up to a class — used in median & ogive.

Relative frequency = frequency / total (useful for proportions).

3. Mean (Measure of Central Tendency)

Ungrouped data

Mean = (Σx) / n

Grouped data (using class marks)

Use class midpoint mi for each class with frequency fi:

Mean = (Σ fi mi) / Σ fi

Short-cut (assumed mean) method to reduce calculations: let A be assumed mean, di = (mi − A)/h, then

Mean = A + h × (Σ fi di / Σ fi)

4. Median

Ungrouped data (odd/even)

Sort data. If n odd, median = middle value. If n even, median = average of two middle values.

Grouped data (continuous)

Median = l + \frac{\left(\frac{N}{2} - cf\right)}{f} \times h

Where l = lower boundary of median class, N = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width.

5. Mode & Modal Class (Grouped Data)

Mode is the value with highest frequency. For grouped data use modal class (class with maximum frequency) and formula:

Mode = l + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h

Where l = lower boundary of modal class, f_1 = frequency of modal class, f_0 = frequency of previous class, f_2 = frequency of next class, h = class width.

If data heavily skewed, mode gives the most frequent outcome; median and mean help understand skewness (mean − mode relationship).

6. Graphs — Histogram & Ogive

Histogram: bar-like graph where area of each rectangle = frequency (width = class width, height = frequency/class width). Use for continuous data.

Ogive (cumulative frequency curve): Plot cumulative frequency vs upper class boundary. Useful to read median, quartiles and percentiles graphically.

To find median from ogive, draw horizontal line at N/2 and read intersection's x-coordinate.

7. Measures of Dispersion (Quick)

  • Range = max − min (simple but sensitive to extremes).
  • Mean Deviation (about mean): MD = (Σ fi |xi − mean|) / N for grouped data (use class marks).
  • Variance & Standard Deviation are usually covered later — note that SD gives spread in same units as mean.
For exams focus: correct use of formulas for grouped & ungrouped data and computing MD for one or two examples.

8. Solved Examples

Example 1 (Ungrouped mean). Data: 4, 7, 8, 6, 10. Mean = (4+7+8+6+10)/5 = 35/5 = 7.

Example 2 (Grouped mean — assumed mean). Classes: 0–10(2),10–20(5),20–30(9),30–40(4). Take class marks 5,15,25,35. Mean = (2×5 + 5×15 + 9×25 + 4×35)/20 = (10 + 75 + 225 + 140)/20 = 450/20 = 22.5.

Example 3 (Median from grouped data). Classes: 0–10(3),10–20(5),20–30(12),30–40(10). N=30. Median class is 20–30 (cf before = 8). Using formula with l=20, h=10, f=12, cf=8: Median = 20 + ((15 − 8)/12)×10 = 20 + (7/12)×10 ≈ 25.83.

Example 4 (Mode grouped). Classes: 0–10(4),10–20(8),20–30(15),30–40(7). Modal class = 20–30 (f1=15, f0=8, f2=7), l=20, h=10 ⇒ Mode = 20 + ((15−8)/(2×15 −8 −7))×10 = 20 + (7/(30−15))×10 = 20 + (7/15)×10 ≈ 24.67.

9. Practice Questions

  1. Find mean, median and mode of: 3, 7, 7, 2, 9, 10, 7.
  2. Construct a frequency distribution for continuous data: 12, 15, 18, 21, 24, 29, 30, 34, 37, 39 (use class width 10 starting at 10).
  3. From grouped data classes 0–10(5),10–20(10),20–30(20),30–40(5), find mean (using class marks) and median.
  4. Draw an ogive for cumulative frequencies: classes 0–10(3),10–20(5),20–30(12),30–40(10) and estimate median graphically.

10. Exam Tips & Quick Checklist

  • Always state whether data is grouped or ungrouped before choosing formula.
  • For grouped data use exact class boundaries when required (e.g., 9.5–19.5 if data are integers) to avoid off-by-one in continuous interpretation.
  • Label axes and use class boundaries for histograms and ogives — width matters for heights in histogram.
  • When using assumed mean method, pick A near central class to reduce computation error and simplify arithmetic.