Chapter – 13

Statistics

In this post we have given the detailed notes of class 10 Maths Chapter 13 (Statistics) in English. These notes are useful for the students who are going to appear in class 10 board exams.

Chapter 13: Statistics

Introduction to Statistics

Statistics is the branch of mathematics that deals with the collection, presentation, analysis and interpretation of numerical data. In Class 9, we learnt how to present raw data in the form of tables and graphs (bar graphs, histograms and frequency polygons), and how to find the mean, median and mode of ungrouped data. In this chapter, we extend these ideas to grouped data, i.e., data that is arranged in class intervals.

The mean, median and mode are collectively called measures of central tendency, since they give us a single representative value that describes where most of the data is centred.

Mean of Grouped Data

The mean (or average) of a set of observations is obtained by dividing the sum of the values of all the observations by the total number of observations. When data is grouped into class intervals, we cannot know the exact value of every observation — we only know how many observations (the frequency) fall in each class. To overcome this, we assume that the frequency of each class interval is centred at its mid-point, also called the class mark.

Class mark, xᵢ = (upper class limit + lower class limit) / 2

There are three methods of finding the mean of grouped data:

1. Direct Method
2. Assumed Mean Method
3. Step-deviation Method

1. Direct Method

If x₁, x₂, x₃, …, xₙ are the observations (class marks) with corresponding frequencies f₁, f₂, f₃, …, fₙ, then the mean x̄ is given by:

x̄ = Σfᵢxᵢ / Σfᵢ

where Σfᵢxᵢ is the sum of the products of each class mark with its frequency, and Σfᵢ is the sum of all the frequencies (i.e., the total number of observations, n).

Solved Example: Find the mean of the following distribution of marks obtained by students.

Solution: Finding the class mark (xᵢ) for each interval and then fᵢxᵢ:

x̄ = Σfᵢxᵢ / Σfᵢ = 1860 / 30 = 62

Note: The Direct Method becomes tedious when the values of xᵢ and fᵢ are large. In such cases, the Assumed Mean Method or the Step-deviation Method is preferred.

2. Assumed Mean Method

In this method, we choose one of the xᵢ values (preferably one in the middle) as the assumed mean, denoted by “a”. We then find the deviation, dᵢ, of each xᵢ from a:

dᵢ = xᵢ − a

The mean is then given by:

x̄ = a + (Σfᵢdᵢ / Σfᵢ)

3. Step-deviation Method

If all the class sizes (widths) are equal (say, h), the calculation can be simplified further by dividing the deviations dᵢ by h:

uᵢ = (xᵢ − a) / h

The mean is then given by:

x̄ = a + h × (Σfᵢuᵢ / Σfᵢ)

Solved Example (all three methods on the same data): The following table shows the number of plants in 20 houses of a locality. Find the mean number of plants per house.

By Direct Method:

x̄ = 162 / 20 = 8.1 plants

By Assumed Mean Method (taking a = 7, the class mark of 6–8):

dᵢ values: −6, −4, −2, 0, 2, 4, 6; corresponding fᵢdᵢ: −6, −8, −2, 0, 12, 8, 18

Σfᵢdᵢ = −6 − 8 − 2 + 0 + 12 + 8 + 18 = 22

x̄ = a + Σfᵢdᵢ/Σfᵢ = 7 + 22/20 = 7 + 1.1 = 8.1

By Step-deviation Method (a = 7, h = 2):

uᵢ values: −3, −2, −1, 0, 1, 2, 3; corresponding fᵢuᵢ: −3, −4, −1, 0, 6, 4, 9

Σfᵢuᵢ = −3 − 4 − 1 + 0 + 6 + 4 + 9 = 11

x̄ = a + h × (Σfᵢuᵢ/Σfᵢ) = 7 + 2 × (11/20) = 7 + 1.1 = 8.1

Notice that all three methods give the same mean, 8.1, since they are mathematically equivalent — only the arithmetic is simplified.

Another Solved Example (Step-deviation Method): The following distribution shows the daily expenditure on food of 25 households in a locality. Find the mean daily expenditure on food.

Solution: Taking a = 225 (class mark of 200–250) and h = 50:

x̄ = a + h × (Σfᵢuᵢ/Σfᵢ) = 225 + 50 × (−7/25) = 225 − 14 = ₹211

Mode of Grouped Data

The mode is the value among the observations which occurs most frequently, i.e., the value having the maximum frequency.

Solved Example (ungrouped data): The number of wickets taken by a bowler in 10 cricket matches is: 2, 6, 4, 5, 0, 2, 1, 3, 2, 3. Find the mode.

Solution: Arranging the data in a frequency table, we see that 2 occurs the maximum number of times (3 times). Hence, the mode = 2.

Modal Class

In a grouped frequency distribution, it is not possible to find the mode simply by looking at the frequencies — we can only find the class with the maximum frequency. This class is called the modal class, and the mode is a value inside this class, found using the formula:

Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h

where:

• l = lower limit of the modal class
• h = size of the class interval (assuming all class sizes are equal)
• f₁ = frequency of the modal class
• f₀ = frequency of the class preceding (before) the modal class
• f₂ = frequency of the class succeeding (after) the modal class

Solved Example: A survey conducted on 20 households in a locality gave the following distribution of the number of family members:

Solution: The maximum frequency is 8, corresponding to the class 3–5. So, the modal class is 3–5.

l = 3, h = 2, f₁ = 8, f₀ = 7, f₂ = 2

Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h = 3 + [(8 − 7)/(16 − 7 − 2)] × 2 = 3 + (1/7) × 2 = 3 + 0.286 = 3.286

So, the mode of this data is 3.286.

Median of Grouped Data

The median is a measure of central tendency that gives the value of the middle-most observation in the data, when the observations are arranged in ascending or descending order.

For ungrouped data, we first arrange the n observations in ascending order. If n is odd, the median is the value of the ((n + 1)/2)th observation. If n is even, the median is the mean (average) of the (n/2)th and the ((n/2) + 1)th observations.

Cumulative Frequency

For grouped data, we first need to find the class in which the median lies (called the median class). For this, we calculate the cumulative frequency (cf) of each class — this is the running total of frequencies up to and including that class.

Median Class

To find the median class, calculate n/2. Then locate the class whose cumulative frequency is greater than (and nearest to) n/2. This is the median class. Once the median class is known, the median is found using the formula:

Median = l + [(n/2 − cf) / f] × h

where:

• l = lower limit of the median class
• n = number of observations
• cf = cumulative frequency of the class preceding the median class
• f = frequency of the median class
• h = class size (assuming class size is equal)

Solved Example: A survey of heights of 51 girls of Class X of a school was conducted, and the following data was obtained:

Find the median height.

Solution: Converting to a class-interval frequency table with cumulative frequencies:

n = 51, so n/2 = 25.5. The class whose cf is just greater than 25.5 is 145–150. So, the median class is 145–150.

l = 145, cf (of the class preceding median class) = 11, f = 18, h = 5

Median = l + [(n/2 − cf)/f] × h = 145 + [(25.5 − 11)/18] × 5 = 145 + (14.5/18) × 5 = 145 + 4.03 = 149.03 cm

This means that about half the girls have heights less than 149.03 cm and half have heights more than 149.03 cm.

Comparing Mean, Median and Mode: A Combined Example

The following distribution gives the monthly consumption of electricity (in units) of 68 consumers of a locality. Find the median, mean and mode of the data, and compare them.

For Median: Cumulative frequencies are 4, 9, 22, 42, 56, 64, 68. n = 68, so n/2 = 34. The class whose cf is just greater than 34 is 125–145 (cf = 42). So the median class is 125–145.

l = 125, cf (of preceding class) = 22, f = 20, h = 20

Median = 125 + [(34 − 22)/20] × 20 = 125 + 12 = 137

For Mode: The maximum frequency is 20, corresponding to the class 125–145. So the modal class is 125–145.

l = 125, f₁ = 20, f₀ = 13, f₂ = 14, h = 20

Mode = 125 + [(20 − 13)/(40 − 13 − 14)] × 20 = 125 + (7/13) × 20 = 125 + 10.77 = 135.76

For Mean (Step-deviation Method, a = 135, h = 20):

Σfᵢuᵢ = 7, Σfᵢ = 68

Mean = 135 + 20 × (7/68) = 135 + 2.06 = 137.06

Comparison: Mean = 137.06, Median = 137, Mode = 135.76. All three measures are close to each other in this data.

Empirical Relationship Between the Three Measures

There is an empirical relationship between the three measures of central tendency:

3 Median = Mode + 2 Mean

This relationship is approximate and holds reasonably well for moderately skewed distributions; it can also be used to estimate one measure when the other two are known.

Key Points to Remember

• Class mark (xᵢ) = (upper limit + lower limit)/2
• Direct Method: x̄ = Σfᵢxᵢ / Σfᵢ
• Assumed Mean Method: x̄ = a + (Σfᵢdᵢ/Σfᵢ), where dᵢ = xᵢ − a
• Step-deviation Method: x̄ = a + h × (Σfᵢuᵢ/Σfᵢ), where uᵢ = (xᵢ − a)/h
• Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, for the modal class (class with the highest frequency)
• Median = l + [(n/2 − cf)/f] × h, for the median class (found using cumulative frequency and n/2)
• Empirical relationship: 3 Median = Mode + 2 Mean
• Mean is affected by extreme values, Median divides the data exactly into two halves, and Mode is the most frequently occurring value — each is useful in different situations.

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