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Statistics: Grouped Data, Mean, Median, Mode, and Ogives

Complete Class 10 Statistics notes covering grouped frequency distributions, class marks, mean by direct and assumed-mean methods, median, mode, and ogive-based interpretation.

What This Chapter Covers

This chapter teaches how to summarise large sets of data using mean, median, and mode. In Class 10, the main focus is on grouped data, where observations are collected in class intervals.

You also learn how cumulative frequency helps us locate the median class, and how an ogive lets us estimate the median graphically.

Quick Formula Strip

\bar{x} = \frac{\sum f_i x_i}{\sum f_i}

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

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

x_i = \frac{\text{lower limit} + \text{upper limit}}{2}

Chapter Roadmap

What Statistics Studies Frequency Distribution Grouped Data and Class Marks Mean of Grouped Data Median of Grouped Data Mode of Grouped Data Ogives and Reading the Median

What Statistics Studies

Statistics turns raw observations into a readable summary.

In Class 10, statistics is about organising data, spotting patterns, and summarising a large collection of values with a few meaningful numbers.

The three main measures of central tendency in this chapter are mean, median, and mode. The chapter also uses cumulative frequency and graphs called ogives to interpret grouped data.

\text{Mean} = \frac{\text{sum of observations}}{\text{number of observations}}

\text{Median} = \text{middle value of ordered data}

\text{Mode} = \text{most frequent observation}

Board Exam Lens

Class 10 Statistics is not about advanced probability language. It is mainly about grouped frequency tables, mean, median, mode, and ogives.

Pick the right measure quickly

Which average is the most useful when you want the most common shirt size sold in a shop?

Show solution

The question asks for the value that occurs most often.

That is exactly what mode measures.

So the best answer is mode.

Practice Prompt

For a class test, what would mean, median, and mode each tell you in plain language?

Frequency Distribution

A frequency table compresses repeated values into a clean structure.

When many observations repeat, we write each value once and record how many times it occurs. That count is called the frequency.

If the data set is large, we often group values into class intervals such as 0-10, 10-20, 20-30, and so on. This is called grouped data.

\sum f = N

The total of all frequencies is the number of observations.

Quick Habit

Before solving any grouped-data problem, identify the class interval, frequency column, and total frequency N. That prevents most setup mistakes.

Read the frequency total

A table has frequencies 3, 5, 7, and 9. How many observations are there in total?

Show solution

Add all frequencies:

N = 3 + 5 + 7 + 9 = 24

So the data set has 24 observations.

Practice Prompt

Why is the sum of frequencies important before finding median or drawing an ogive?

Grouped Data and Class Marks

The class mark is the midpoint used in the mean formulas.

For grouped data, we do not know every exact observation inside a class interval, so we represent the class by its midpoint. This midpoint is called the class mark.

If a class interval is 20-30, then its class mark is 25. This value is used in direct and assumed-mean methods.

x_i = \frac{\text{upper limit} + \text{lower limit}}{2}

Find a class mark

Find the class mark of the interval 35-45.

Show solution

Use the midpoint formula:

x_i = \frac{35+45}{2} = \frac{80}{2} = 40

So the class mark is 40.

Practice Prompt

Find the class marks of 10-20, 20-30, 30-40, and 40-50.

Mean of Grouped Data

The mean is a weighted average using the class marks.

For grouped data, the direct method uses class marks and frequencies. Each class mark is multiplied by its frequency, and then we divide the total by the total frequency.

When numbers are large, the assumed mean or step-deviation method reduces arithmetic. The concept stays the same: we are still computing a weighted average.

\bar{x} = \frac{\sum f_i x_i}{\sum f_i}

\bar{x} = a + \frac{\sum f_i d_i}{\sum f_i}

Assumed mean method where d_i = x_i - a.

\bar{x} = a + h\frac{\sum f_i u_i}{\sum f_i}

Step-deviation method where u_i = \frac{x_i-a}{h}.

Fast Choice

If class marks are neat and small, direct method is fine. If class marks are large but equally spaced, assumed mean or step-deviation is faster.

Mean by direct method

For class marks 5, 15, 25 with frequencies 2, 3, 5, find the mean.

Show solution

Compute

\sum f_i x_i = 2(5)+3(15)+5(25)=10+45+125=180

Total frequency

\sum f_i = 2+3+5=10

\bar{x}=\frac{180}{10}=18

Hence the mean is 18.

Practice Prompt

Using class marks 10, 20, 30, 40 and frequencies 4, 6, 5, 5, find the mean.

Median of Grouped Data

Median is the central value located through cumulative frequency.

For grouped data, we first compute cumulative frequencies and find $N/2$ . The class whose cumulative frequency first exceeds $N/2$ is called the median class.

After locating the median class, we use the standard median formula for grouped data.

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

l = \text{lower boundary of median class},\ cf = \text{cumulative frequency before median class}

Identify the median class

The cumulative frequencies are 4, 11, 19, 28, 35. Find the position that decides the median class.

Show solution

Total frequency

N=35

\frac{N}{2}=17.5

The first cumulative frequency greater than 17.5 is 19.

So the corresponding class is the median class.

Practice Prompt

Why do we use cumulative frequency instead of ordinary frequency while finding the grouped median?

Mode of Grouped Data

Mode identifies the class with the highest concentration of observations.

The class with the highest frequency is called the modal class. We then refine the answer using the grouped-data mode formula.

Mode is very useful when the question asks for the most common size, score band, price band, or repeated category.

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

f_1 = \text{frequency of modal class},\ f_0 = \text{previous frequency},\ f_2 = \text{next frequency}

Modal Class

Always identify the modal class first. If you choose the wrong class, the whole mode calculation collapses even if the formula is correct.

Spot the modal class first

Frequencies of five classes are 4, 9, 12, 7, 3. Which class is the modal class?

Show solution

The highest frequency is 12.

So the class corresponding to frequency 12 is the modal class.

Practice Prompt

If the highest frequency is in the 30-40 class, which values become f1, f0, and f2?

Ogives and Reading the Median

An ogive is a cumulative-frequency curve used to estimate the median visually.

A less-than ogive is drawn by plotting upper class boundaries against cumulative frequencies. A more-than ogive uses lower class boundaries against more-than cumulative frequencies.

The median can be read graphically by marking $N/2$ on the frequency axis, drawing across to the ogive, and then dropping a vertical line to the x-axis.

\text{Median position on ogive} = \frac{N}{2}

Ogive interpretation idea

If a grouped table has total frequency 60, at what y-value do you start when finding the median on the ogive?

Show solution

For an ogive-based median, start with

\frac{N}{2}

\frac{60}{2}=30

You mark 30 on the y-axis, move horizontally to the ogive, and then drop vertically to read the median.

Practice Prompt

What is the first number you compute before locating the median on an ogive?

Revise and Practise

After reading the notes, move to reveal-style practice so you can test the grouped-data formulas, locate median class correctly, and avoid confusion between mean, median, and mode.

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