Class 10 Maths Statistics Notes

13.1 Introduction to Statistics

Statistics is the branch of mathematics that deals with the collection, organization, analysis, and interpretation of data. In real life, data is collected from surveys, experiments, or observations. To make sense of large amounts of data, we use measures like mean, median, and mode.

• Raw Data: Data collected in its original form.
• Grouped Data: Data is arranged in intervals (classes) for easier analysis.

In this chapter, we focus on grouped data and how to calculate mean, mode, and median using formulas.

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13.2 Mean of Grouped Data

The mean (average) is the sum of all observations divided by the total number of observations. For grouped data, we use the class midpoints ($x_i$xi​) and frequencies ($f_i$fi​):$$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i}$$Mean=∑fi​∑fi​xi​​

Where:

• $f_i$fi​ = frequency of the $i$i-th class
• $x_i$xi​ = midpoint of the $i$i-th class
• $\sum f_i x_i$∑fi​xi​ = sum of frequency × midpoint

Example:
Consider the following frequency distribution:

Class	Frequency (f)
0–10	5
10–20	8
20–30	12
30–40	5

Step 1: Find midpoints ($x_i$xi​)$$x_i = \frac{\text{Lower limit + Upper limit}}{2}$$xi​=2Lower limit + Upper limit​

Class	Midpoint $x_i$xi​	Frequency $f_i$fi​	$f_i x_i$fi​xi​
0–10	5	5	25
10–20	15	8	120
20–30	25	12	300
30–40	35	5	175

$$\text{Mean} = \frac{\sum f_i x_i}{\sum f_i} = \frac{25 + 120 + 300 + 175}{5 + 8 + 12 + 5} = \frac{620}{30} \approx 20.67$$Mean=∑fi​∑fi​xi​​=5+8+12+525+120+300+175​=30620​≈20.67

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13.3 Mode of Grouped Data

The mode is the value that occurs most frequently. For grouped data, we use the modal class (the class with the highest frequency) and the formula:$$\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h$$Mode=l+2f1​−f0​−f2​f1​−f0​​⋅h

Where:

• $l$l = lower limit of modal class
• $h$h = class width
• $f_1$f1​ = frequency of modal class
• $f_0$f0​ = frequency of previous class
• $f_2$f2​ = frequency of next class

Example:
Using the previous data, the modal class is 20–30 (frequency 12).$$\text{Mode} = 20 + \frac{12-8}{2(12) – 8 – 5} \cdot 10 = 20 + \frac{4}{24-13} \cdot 10 = 20 + \frac{4}{11} \cdot 10 \approx 23.64$$Mode=20+2(12)−8−512−8​⋅10=20+24−134​⋅10=20+114​⋅10≈23.64

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13.4 Median of Grouped Data

The median is the middle value that divides the data into two equal parts. For grouped data, we use the median class and the formula:$$\text{Median} = l + \frac{\frac{N}{2} – CF}{f} \cdot h$$Median=l+f2N​−CF​⋅h

Where:

• $l$l = lower limit of median class
• $N$N = total frequency ($\sum f_i$∑fi​)
• $CF$CF = cumulative frequency of class before median class
• $f$f = frequency of median class
• $h$h = class width

Example:
Total frequency $N = 30$N=30. $\frac{N}{2} = 15$2N​=15.

Cumulative frequencies:

• 0–10 → 5
• 10–20 → 13
• 20–30 → 25

Median class = 20–30 (since cumulative frequency just before it is 13 < 15 ≤ 25).$$\text{Median} = 20 + \frac{15-13}{12} \cdot 10 = 20 + \frac{2}{12} \cdot 10 = 20 + 1.67 \approx 21.67$$Median=20+1215−13​⋅10=20+122​⋅10=20+1.67≈21.67

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13.5 Summary

• Mean gives the average value of a dataset.
• Mode gives the most frequent value in the dataset.
• Median gives the middle value dividing the data into two equal halves.
• For grouped data, we use class midpoints, cumulative frequencies, and formulas to calculate mean, median, and mode.

MCQs Based on the “Statistics” Chapter:

1. What is the mean of a dataset?

a) The most frequent value
b) The middle value
c) The average of all observations
d) The difference between highest and lowest value

Answer: c) The average of all observations

2. Which class is called the modal class in grouped data?

a) The first class
b) The class with the highest frequency
c) The class with the lowest frequency
d) The class containing the median

Answer: b) The class with the highest frequency

3. To find the median of grouped data, which information is required?

a) Class width, lower limit of median class, cumulative frequency before median class, frequency of median class
b) Only frequency of median class
c) Only class width
d) Only lower limit

Answer: a) Class width, lower limit of median class, cumulative frequency before median class, frequency of median class

4. The mean of the following grouped data is:

Class	Frequency
0–5	3
5–10	7
10–15	10
15–20	5

a) 10.5
b) 11.0
c) 12.0
d) 12.5

Answer: a) 10.5

5. Which of the following is NOT a measure of central tendency?

a) Mean
b) Median
c) Mode
d) Range

Answer: d) Range