CH – 13 STATISTICS
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Chapter Overview
In this chapter, you will learn how to analyze and summarize large sets of data using numerical measures of central tendency.
- Understanding grouped data: Grouping large data into class intervals to make it meaningful and organized.
- Class intervals & Frequency: Defining continuous data ranges and counting observation occurrences in each range.
- Class marks: Finding the middle value of each class interval to represent the class.
- Mean of grouped data: Calculating the average value using Direct, Assumed Mean, and Step-Deviation methods.
- Cumulative frequency: Adding frequencies continuously to find running totals for data points.
- Median of grouped data: Locating the middle-most value of a grouped dataset.
- Mode of grouped data: Finding the value that occurs most frequently in grouped data.
- Using the correct method: Choosing the right calculation method based on data size and interval structure.
🎯 Board Exam Focus
Mastering the following core skills will help you solve exam problems accurately:
- Reading frequency distribution tables: Extracting lower limits, upper limits, and frequencies correctly.
- Finding class marks correctly: Computing mid-points without calculation errors.
- Finding the mean: Mastering Direct, Assumed Mean, and Step-Deviation procedures.
- Finding the median: Correctly evaluating cumulative frequencies and applying the formula.
- Finding the mode: Identifying modal parameters carefully.
- Making cumulative frequency tables: Adding frequencies sequentially without missing any term.
- Identifying the median class: Finding the first class where cumulative frequency is greater than or equal to N ÷ 2.
- Identifying the modal class: Locating the class interval with the highest frequency.
- Substituting values carefully into formulas: Double-checking values of l, cf, f, f₀, f₁, f₂, and h before solving.
- Avoiding calculation mistakes: Paying close attention to negative signs and division operations.
Most Important Concepts
What is Statistics?
Statistics is a branch of mathematics that helps us collect, organize, analyze, and interpret data. In very simple terms, statistics helps us make sense of numbers.
For example, if a teacher records the test marks of 30 students, statistics helps the teacher find the average score of the class instead of looking at 30 individual marks one by one.
What is Grouped Data?
When data is large, reading individual numbers becomes confusing. Grouped data means condensing large numbers of observations into smaller groups called class intervals.
For example, instead of writing marks of 50 students individually, we group them as 0–10, 10–20, 20–30, and count how many students fall into each group.
VISUAL REPRESENTATION 1 — GROUPED DATA
Plaintext
+----------------+--------------------+
| Class Interval | Frequency (fᵢ) |
+----------------+--------------------+
| 10 - 25 | 2 |
| 25 - 40 | 3 |
| 40 - 55 | 7 |
| 55 - 70 | 6 |
+----------------+--------------------+
| Total | Σfᵢ = 18 |
+----------------+--------------------+
Important Words You Must Know
Class Interval
A class interval is a group or range of values used to organize data. Examples of class intervals:
- 10–20
- 20–30
- 30–40
In 10–20, 10 is the lower class limit and 20 is the upper class limit.
Frequency
Frequency tells us how many times a particular observation or data item occurs in a class interval. For example, if 7 students scored marks between 40 and 55, the frequency of the class interval 40–55 is 7.
Class Mark
The class mark is the middle value (or mid-point) of a class interval. It acts as a representative for all values in that group.
Class Mark = (Upper Class Limit + Lower Class Limit) ÷ 2
Example: For class interval 10–25: Class Mark = (25 + 10) ÷ 2 = 35 ÷ 2 = 17.5
VISUAL REPRESENTATION 2 — CLASS MARK
Plaintext
[ Class Interval: 10 - 25 ]
↓
Lower Limit = 10 , Upper Limit = 25
↓
Add limits: (10 + 25) = 35
↓
Divide by 2: 35 ÷ 2
↓
[ Class Mark (xᵢ) = 17.5 ]
📌 Mean of Grouped Data
The mean is the average of all given observations. It represents the single central value around which the data is distributed.
Direct Method
Use this method when the numbers for class mark (xᵢ) and frequency (fᵢ) are small.
Formula:
Mean (x̄) = Σfᵢxᵢ ÷ Σfᵢ
Where:
- xᵢ = Class mark of the class interval
- fᵢ = Frequency of the class interval
- fᵢxᵢ = Frequency × Class mark
- Σfᵢxᵢ = Sum of all fᵢxᵢ products
- Σfᵢ = Sum of all frequencies
Steps for Direct Method:
- Find the class mark (xᵢ) for each class interval using (Upper limit + Lower limit) ÷ 2.
- Multiply each frequency (fᵢ) by its class mark (xᵢ) to get fᵢxᵢ.
- Add all values in the fᵢxᵢ column to get Σfᵢxᵢ.
- Add all frequencies together to get Σfᵢ.
- Divide Σfᵢxᵢ by Σfᵢ.
Example:
Plaintext
+----------------+----------------+----------------+------------------+
| Class Interval | Frequency (fᵢ) | Class Mark(xᵢ) | fᵢxᵢ |
+----------------+----------------+----------------+------------------+
| 10 - 25 | 2 | 17.5 | 2 × 17.5 = 35 |
| 25 - 40 | 3 | 32.5 | 3 × 32.5 = 97.5 |
| 40 - 55 | 7 | 47.5 | 7 × 47.5 = 332.5 |
+----------------+----------------+----------------+------------------+
| Total | Σfᵢ = 12 | | Σfᵢxᵢ = 465.0 |
+----------------+----------------+----------------+------------------+
Mean = Σfᵢxᵢ ÷ Σfᵢ = 465 ÷ 12 = 38.75
VISUAL REPRESENTATION 3 — MEAN CALCULATION FLOW
Plaintext
CLASS INTERVAL
↓
FIND CLASS MARK (xᵢ)
↓
MULTIPLY (fᵢ × xᵢ)
↓
ADD ALL fᵢxᵢ VALUES → (Σfᵢxᵢ)
↓
DIVIDE BY TOTAL FREQUENCY (Σfᵢ)
↓
[ MEAN (x̄) ]
🧠 Easy Way to Remember Mean
Mean = Total value of observations ÷ Total number of observations
📌 Assumed Mean Method
When xᵢ and fᵢ are large, multiplying them directly takes time and leads to errors. The Assumed Mean Method reduces big numbers into smaller numbers, making calculation easy.
Terms Used:
- Assumed Mean (A): A value chosen as the middle entry from the class marks (xᵢ).
- Deviation (dᵢ): Difference between each class mark and assumed mean: dᵢ = xᵢ − A.
Formula:
Mean (x̄) = A + (Σfᵢdᵢ ÷ Σfᵢ)
Example: Let Assumed Mean (A) = 47.5.
Plaintext
+----------------+----------------+---------------+---------------+---------------+
| Class Interval | Frequency (fᵢ) | Class Mark(xᵢ)| dᵢ = xᵢ - A | fᵢdᵢ |
+----------------+----------------+---------------+---------------+---------------+
| 10 - 25 | 2 | 17.5 | 17.5-47.5=-30 | 2 × -30 =-60 |
| 25 - 40 | 3 | 32.5 | 32.5-47.5=-15 | 3 × -15 =-45 |
| 40 - 55 | 7 | 47.5 | 47.5-47.5= 0 | 7 × 0 = 0 |
| 55 - 70 | 6 | 62.5 | 62.5-47.5= 15 | 6 × 15 = 90 |
+----------------+----------------+---------------+---------------+---------------+
| Total | Σfᵢ = 18 | | | Σfᵢdᵢ = -15 |
+----------------+----------------+---------------+---------------+---------------+
Mean = A + (Σfᵢdᵢ ÷ Σfᵢ) = 47.5 + (-15 ÷ 18) = 47.5 − 0.83 = 46.67
📌 Step-Deviation Method
When class intervals have an equal width (h), and the deviations (dᵢ) share a common factor, the Step-Deviation method simplifies calculations even further.
Terms Used:
- Class size (h): Upper Limit − Lower Limit
- uᵢ:uᵢ = (xᵢ − A) ÷ h
Formula:
Mean (x̄) = A + [(Σfᵢuᵢ ÷ Σfᵢ) × h]
Example: Let A = 47.5 and class size h = 15.
Plaintext
+----------------+----------------+---------------+---------------+------------------+--------------+
| Class Interval | Frequency (fᵢ) | Class Mark(xᵢ)| dᵢ = xᵢ - A | uᵢ = (xᵢ - A)/15 | fᵢuᵢ |
+----------------+----------------+---------------+---------------+------------------+--------------+
| 10 - 25 | 2 | 17.5 | -30 | -2 | 2 × -2 = -4 |
| 25 - 40 | 3 | 32.5 | -15 | -1 | 3 × -1 = -3 |
| 40 - 55 | 7 | 47.5 | 0 | 0 | 7 × 0 = 0 |
| 55 - 70 | 6 | 62.5 | 15 | 1 | 6 × 1 = 6 |
+----------------+----------------+---------------+---------------+------------------+--------------+
| Total | Σfᵢ = 18 | | | | Σfᵢuᵢ = -1 |
+----------------+----------------+---------------+---------------+------------------+--------------+
Mean = A + [(Σfᵢuᵢ ÷ Σfᵢ) × h] Mean = 47.5 + [(-1 ÷ 18) × 15] = 47.5 + [-15 ÷ 18] = 47.5 − 0.83 = 46.67
🧠 How to Choose the Mean Method?
- Direct Method: Use when frequencies and class marks are small numbers (easy mental math).
- Assumed Mean Method: Use when numbers are large and multiplication becomes tedious.
- Step-Deviation Method: Best choice when class sizes are equal and deviations have a clear common divisor, giving tiny single-digit values to multiply.
🧠 Cumulative Frequency
Cumulative frequency (cf) is calculated by continuously adding frequencies step-by-step as you move down the table.
Example:
- First frequency = 5 → cf = 5
- Second frequency = 3 → cf = 5 + 3 = 8
- Third frequency = 4 → cf = 8 + 4 = 12
VISUAL REPRESENTATION 4 — CUMULATIVE FREQUENCY
Plaintext
Frequency (f) Cumulative Frequency (cf)
5 --------------> 5
3 --------------> 5 + 3 = 8
4 --------------> 8 + 4 = 12
7 --------------> 12 + 7 = 19
📌 Median of Grouped Data
The median represents the middle value of a given dataset.
Step 1: Find Total Frequency
Calculate N = Σfᵢ.
Step 2: Find N ÷ 2
Divide total frequency by 2.
Step 3: Find the Median Class
Locate the class interval whose cumulative frequency (cf) is greater than and nearest to N ÷ 2. This class interval is the Median Class.
VISUAL REPRESENTATION 5 — FINDING THE MEDIAN CLASS
Plaintext
Calculate total frequency (N)
↓
Compute (N ÷ 2)
↓
Look down the Cumulative Frequency (cf) column
↓
Locate the first cf value that is ≥ (N ÷ 2)
↓
[ THAT CLASS IS THE MEDIAN CLASS ]
Median Formula
Median = l + [ ((N ÷ 2) − cf) ÷ f ] × h
Where:
- l = Lower boundary of the median class
- N = Total frequency (Σfᵢ)
- cf = Cumulative frequency of the class preceding (before) the median class
- f = Frequency of the median class itself
- h = Class size
Example:
Plaintext
+----------------+----------------+---------------------------+
| Class Interval | Frequency (f) | Cumulative Frequency (cf) |
+----------------+----------------+---------------------------+
| 0 - 10 | 5 | 5 |
| 10 - 20 | 3 | 8 |
| 20 - 30 | 4 | 12 |
| 30 - 40 | 3 | 15 |
| 40 - 50 | 3 | 18 |
+----------------+----------------+---------------------------+
| Total | N = 18 | |
+----------------+----------------+---------------------------+
- N = 18, so N ÷ 2 = 9.
- Cumulative frequency just greater than 9 is 12 (in class 20–30).
- Therefore, Median Class = 20–30.
- Values: l = 20, cf = 8 (cumulative frequency of class before median class), f = 4, h = 10.
Median = 20 + [ ((9 − 8) ÷ 4) × 10 ] Median = 20 + [ (1 ÷ 4) × 10 ] = 20 + 2.5 = 22.5
📌 Mode of Grouped Data
Mode is the value that occurs most frequently in a dataset. In grouped data, the class with the highest frequency is called the Modal Class.
VISUAL REPRESENTATION 6 — MODE
Plaintext
Class Interval Frequency
10 - 20 4
20 - 30 15 <--- HIGHEST FREQUENCY
30 - 40 7
↓
[ MODAL CLASS = 20 - 30 ]
🧠 How to Identify the Modal Class?
Look at the frequency column of all class intervals. The interval having the largest number as its frequency is your modal class.
Mode Formula
Mode = l + [ (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) ] × h
Where:
- l = Lower boundary of the modal class
- 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
- h = Class size
Example:
Plaintext
+----------------+----------------+
| Class Interval | Frequency (f) |
+----------------+----------------+
| 1 - 3 | 7 |
| 3 - 5 | 8 <--- | (Modal Class: 3 - 5)
| 5 - 7 | 2 |
+----------------+----------------+
- Highest frequency is 8, so Modal Class = 3–5.
- l = 3, f₁ = 8, f₀ = 7, f₂ = 2, h = 2.
Mode = 3 + [ (8 − 7) ÷ (2(8) − 7 − 2) ] × 2 Mode = 3 + [ 1 ÷ (16 − 9) ] × 2 = 3 + (1 ÷ 7) × 2 = 3 + 0.286 = 3.286
📌 Important Relationship Between Mean, Median and Mode
There is an empirical relationship connecting the three measures of central tendency:
3 × Median = Mode + 2 × Mean
Alternatively written as:
Mode = 3 × Median − 2 × Mean
🧠 Mean vs Median vs Mode
VISUAL REPRESENTATION 7 — MEAN, MEDIAN AND MODE
Plaintext
+-----------+-----------------------+----------------------------------+
| Measure | Meaning | Quick Trick |
+-----------+-----------------------+----------------------------------+
| MEAN | Average value | Sum ÷ Total Count |
| MEDIAN | Middle-most value | Position at N ÷ 2 |
| MODE | Most frequent value | Peak / Highest Frequency Class |
+-----------+-----------------------+----------------------------------+
🧠 Concepts Students Find Difficult
- Finding class marks: Confusing limits and adding incorrectly before dividing.
- Choosing the assumed mean (A): Selecting A from the frequency column instead of the class mark column.
- Calculating deviations (dᵢ): Making sign errors when calculating xᵢ − A.
- Finding cumulative frequency: Adding numbers out of order across rows.
- Identifying the median class: Looking for N ÷ 2 in the frequency column instead of cumulative frequency column.
- Understanding cf in Median formula: Taking the cf of the median class itself instead of taking the cf of the class before it.
- Identifying f₀, f₁, f₂ in Mode formula: Swapping the positions of frequencies preceding and succeeding the modal class.
⚠️ Common Mistakes Students Make
- Calculating Class Mark Incorrectly
- Wrong: Subtracting limits instead of adding them: (Upper − Lower) ÷ 2.
- Correction: Always add both limits first: (Upper Limit + Lower Limit) ÷ 2.
- Selecting Assumed Mean from Frequency Column
- Wrong: Picking value ‘A’ from the frequency column (fᵢ).
- Correction: Always choose ‘A’ from the middle of the class mark column (xᵢ).
- Confusing Cumulative Frequency with Frequency
- Wrong: Applying the median formula using frequency values as cumulative frequency.
- Correction: Create a separate cumulative frequency column by adding frequencies step-by-step before using the median formula.
- Selecting Wrong Cumulative Frequency (cf) for Median Formula
- Wrong: Using the cumulative frequency of the median class.
- Correction: Use the cumulative frequency of the class preceding (just before) the median class.
- Picking the Wrong Frequency (f) in Median Formula
- Wrong: Taking the frequency of the preceding class.
- Correction: Take the frequency of the median class itself.
- Mixing up f₀, f₁, and f₂ in the Mode Formula
- Wrong: Confusing order of frequencies around the modal class.
- Correction: Remember the order in a table: f₀ (class before), f₁ (modal class), f₂ (class after).
- Sign Errors in Assumed Mean Deviations
- Wrong: Writing negative deviations as positive when xᵢ < A.
- Correction: Always check signs carefully: when xᵢ is smaller than A, dᵢ MUST be negative.
- Forgetting Class Size (h) Multiplication
- Wrong: Forgetting to multiply by h at the end of median and mode formulas.
- Correction: Complete the formula inside the bracket first, then multiply by h, and finally add lower limit l.
🔥 Priority Revision
⭐⭐⭐ HIGH-PRIORITY PRACTICE
- Mean by Direct Method & Assumed Mean Method.
- Median calculation for grouped frequency distribution.
- Mode calculation for grouped frequency distribution.
- Finding missing frequencies when Mean or Median is given.
⭐⭐ IMPORTANT PRACTICE
- Mean by Step-Deviation Method.
- Converting cumulative frequency distribution tables (“less than” type) into continuous class intervals.
- Empirical relation between Mean, Median, and Mode.
⭐ ADDITIONAL REVISION
- Grouped data with non-uniform class widths.
- Comparison and interpretation of Mean, Median, and Mode for a given dataset.
⚡ Statistics Quick Revision Sheet
- Frequency (fᵢ): Number of observations in a class interval.
- Class Mark (xᵢ): (Upper Limit + Lower Limit) ÷ 2.
- Class Size (h): Upper Limit − Lower Limit.
- Direct Mean Formula: x̄ = Σfᵢxᵢ ÷ Σfᵢ
- Assumed Mean Formula: x̄ = A + (Σfᵢdᵢ ÷ Σfᵢ), where dᵢ = xᵢ − A
- Step-Deviation Formula: x̄ = A + [(Σfᵢuᵢ ÷ Σfᵢ) × h], where uᵢ = (xᵢ − A) ÷ h
- Median Class: Class interval where cumulative frequency first becomes ≥ N ÷ 2.
- Median Formula: Median = l + [ ((N ÷ 2) − cf) ÷ f ] × h
- Modal Class: Class interval with the highest frequency.
- Mode Formula: Mode = l + [ (f₁ − f₀) ÷ (2f₁ − f₀ − f₂) ] × h
- Empirical Relationship: Mode = 3 × Median − 2 × Mean
📝 Test Yourself
- Find the class marks for the class intervals 15–35, 35–55, and 55–75.
- If the lower limit of a class interval is 40 and its class mark is 50, find its upper limit.
- Calculate the mean of the following distribution using the Direct Method:
- Class Intervals: 0–10, 10–20, 20–30, 30–40
- Frequencies: 4, 6, 10, 5
- Find the mean of the given data using the Assumed Mean Method:
- Class Intervals: 100–150, 150–200, 200–250, 250–300
- Frequencies: 12, 18, 14, 6
- Calculate the mean using the Step-Deviation Method:
- Class Intervals: 50–60, 60–70, 70–80, 80–90
- Frequencies: 8, 12, 15, 5
- Construct the “less than” cumulative frequency distribution table for:
- Class Intervals: 0–5, 5–10, 10–15, 15–20
- Frequencies: 3, 7, 11, 4
- Identify the median class for the following frequency distribution:
- Class Intervals: 10–20, 20–30, 30–40, 40–50
- Frequencies: 5, 8, 12, 5
- Find the median of the data given below:
- Class Intervals: 0–20, 20–40, 40–60, 60–80
- Frequencies: 6, 10, 8, 4
- Find the mode of the following distribution:
- Class Intervals: 10–25, 25–40, 40–55, 55–70
- Frequencies: 2, 7, 15, 6
- If the mean of a frequency distribution is 25 and its median is 26, calculate its mode using the empirical formula.
🎯 Before Your Exam
☐ I understand grouped data.
☐ I can find class marks.
☐ I know how to calculate the mean using the direct method.
☐ I understand the assumed mean method.
☐ I understand the step-deviation method.
☐ I can prepare a cumulative frequency table correctly.
☐ I can identify the median class.
☐ I can calculate the median of grouped data.
☐ I can identify the modal class.
☐ I can calculate the mode of grouped data.
Stay calm, read each table carefully, and you will ace your mathematics exam!