Class 10 Mathematics Notes Chapter 14 (Statistics) – Mathematics Book

Alright class, let's focus on Chapter 14, Statistics. This is a crucial chapter, not just for your board exams but also frequently tested in various government recruitment exams. Understanding how to summarise and interpret data is a fundamental skill. We'll cover the measures of central tendency: Mean, Median, and Mode for grouped data.

Chapter 14: Statistics - Detailed Notes for Government Exam Preparation

1. Introduction:
Statistics deals with the collection, organisation, analysis, interpretation, and presentation of data. In this chapter, we focus on finding a 'central value' or 'representative value' for a given dataset, known as Measures of Central Tendency. For Class 10 level and competitive exams, we primarily deal with grouped data (data organised into class intervals).

2. Key Concepts & Terminology:

• Class Interval: A range into which data is grouped (e.g., 10-20, 20-30).
• Class Size (h): The difference between the upper class limit and the lower class limit (e.g., for 10-20, h = 20 - 10 = 10). Assumed to be uniform unless stated otherwise.
• Class Mark (xᵢ): The midpoint of a class interval. xᵢ = (Upper Class Limit + Lower Class Limit) / 2. This value represents the entire class for calculation purposes.
• Frequency (fᵢ): The number of observations falling within a particular class interval.
• Cumulative Frequency (cf): The sum of frequencies of all classes up to and including the current class (for 'less than' type) or starting from the current class down to the last class (for 'more than' type). Usually, 'less than' type is used for median calculation.
• Total Number of Observations (N or Σfᵢ): The sum of all frequencies.

3. Mean of Grouped Data (Average)

The mean gives an average value of the dataset. There are three methods to calculate the mean for grouped data:

• (a) Direct Method:

  • Formula: Mean (x̄) = Σ(fᵢ * xᵢ) / Σfᵢ
  • Steps:
    1. Find the class mark (xᵢ) for each class interval.
    2. Multiply the frequency (fᵢ) of each class by its corresponding class mark (xᵢ) to get fᵢxᵢ.
    3. Sum up all fᵢxᵢ values to get Σfᵢxᵢ.
    4. Sum up all frequencies to get Σfᵢ (or N).
    5. Divide Σfᵢxᵢ by Σfᵢ.
  • When to Use: Suitable when the values of fᵢ and xᵢ are relatively small, making calculations manageable.

• (b) Assumed Mean Method (Shortcut Method):

  • Formula: Mean (x̄) = a + (Σ(fᵢ * dᵢ) / Σfᵢ)
  • Where:
    • a = Assumed Mean (usually the class mark of the middle-most class or the class with highest frequency).
    • dᵢ = Deviation of class mark from assumed mean (dᵢ = xᵢ - a).
  • Steps:
    1. Find class marks (xᵢ).
    2. Choose an Assumed Mean (a).
    3. Calculate deviations (dᵢ = xᵢ - a) for each class.
    4. Multiply fᵢ by dᵢ for each class to get fᵢdᵢ.
    5. Sum up all fᵢdᵢ values to get Σfᵢdᵢ.
    6. Sum up all frequencies (Σfᵢ).
    7. Apply the formula.
  • When to Use: Useful when xᵢ values are large, simplifying calculations by working with smaller dᵢ values.

• (c) Step-Deviation Method: (Note: Sometimes removed from syllabus, but good to know for some exams)

  • Formula: Mean (x̄) = a + ( (Σ(fᵢ * uᵢ) / Σfᵢ) * h )
  • Where:
    • a = Assumed Mean.
    • h = Class Size (must be uniform).
    • uᵢ = Step Deviation (uᵢ = (xᵢ - a) / h or uᵢ = dᵢ / h).
  • Steps:
    1. Find class marks (xᵢ).
    2. Choose a and determine h.
    3. Calculate uᵢ = (xᵢ - a) / h for each class.
    4. Multiply fᵢ by uᵢ for each class (fᵢuᵢ).
    5. Sum up all fᵢuᵢ (Σfᵢuᵢ).
    6. Sum up all frequencies (Σfᵢ).
    7. Apply the formula.
  • When to Use: Best when xᵢ values are large AND the class size (h) is uniform. It simplifies calculations the most by using the smallest deviation values (uᵢ).

4. Mode of Grouped Data

The mode is the value that appears most frequently in the dataset. For grouped data, it lies within the 'modal class'.

• Modal Class: The class interval with the highest frequency (f₁).
• Formula: Mode = l + [ (f₁ - f₀) / (2f₁ - f₀ - f₂) ] * h
• Where:
  • l = Lower limit of the modal class.
  • h = Class size (assuming uniform size).
  • 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.
• Steps:
  1. Identify the class with the maximum frequency (this is the modal class).
  2. Identify the values of l, h, f₁, f₀, and f₂.
  3. Substitute these values into the formula and calculate.

5. Median of Grouped Data

The median is the middle value of the dataset when arranged in ascending or descending order. For grouped data, it represents the value such that half of the observations lie below it and half lie above it.

• Steps:
  1. Prepare a cumulative frequency (cf) column (usually 'less than' type).
  2. Calculate N = Σfᵢ (total frequency).
  3. Find N/2.
  4. Identify the Median Class: The class interval whose cumulative frequency (cf) is just greater than or equal to N/2.
  5. Apply the formula:
     Median = l + [ (N/2 - cf) / f ] * h
• Where:
  • l = Lower limit of the median class.
  • N = Total number of observations (Σfᵢ).
  • cf = Cumulative frequency of the class preceding (before) the median class. (Crucial point!)
  • f = Frequency of the median class.
  • h = Class size (assuming uniform size).

6. Empirical Relationship between Mean, Median, and Mode

For moderately skewed (asymmetrical) distributions, there's an approximate relationship between the three measures of central tendency:

• Formula: 3 * Median = Mode + 2 * Mean
• Use: If any two measures are known, the third can be estimated using this formula. This is often asked in objective questions.

7. Graphical Representation - Ogives (Cumulative Frequency Curves)

• Less than Ogive: Plot points with Upper Class Limits on the x-axis and corresponding Less Than Cumulative Frequencies on the y-axis. Join points with a smooth curve. Starts from the lower limit of the first class with cf 0.
• More than Ogive: Plot points with Lower Class Limits on the x-axis and corresponding More Than Cumulative Frequencies on the y-axis. Join points with a smooth curve. Starts from the lower limit of the first class with cf N.
• Finding Median Graphically: The x-coordinate of the point of intersection of the 'Less than Ogive' and the 'More than Ogive' gives the Median of the data. Alternatively, locate N/2 on the y-axis, draw a horizontal line to intersect the 'Less than Ogive', and then draw a vertical line from this intersection point to the x-axis. The value on the x-axis is the Median.

Multiple Choice Questions (MCQs)

1. The class mark of the class interval 15-25 is:
   a) 15
   b) 25
   c) 20
   d) 10

2. Which of the following is NOT a measure of central tendency?
   a) Mean
   b) Median
   c) Mode
   d) Range

3. In the formula for Mode = l + [ (f₁ - f₀) / (2f₁ - f₀ - f₂) ] * h, f₀ represents:
   a) Frequency of the modal class
   b) Frequency of the class preceding the modal class
   c) Frequency of the class succeeding the modal class
   d) Cumulative frequency of the modal class

4. To find the median of grouped data, the cumulative frequency column required is usually of which type?
   a) More than type
   b) Less than type
   c) Both types are always needed
   d) Frequency density type

5. For a frequency distribution, the Mean is 25 and the Median is 26. Using the empirical relationship, the Mode is approximately:
   a) 24
   b) 25.5
   c) 27
   d) 28

6. Consider the following frequency distribution:

   Class	0-10	10-20	20-30	30-40
   Frequency	5	12	15	8
   The modal class is:
   a) 0-10
   b) 10-20
   c) 20-30
   d) 30-40

7. In the formula for Median = l + [ (N/2 - cf) / f ] * h, cf represents the cumulative frequency of:
   a) The median class
   b) The class preceding the median class
   c) The class succeeding the median class
   d) The last class

8. The method used to calculate the mean when class marks (xᵢ) and frequencies (fᵢ) are large is preferably:
   a) Direct Method
   b) Assumed Mean Method or Step-Deviation Method
   c) Mode calculation method
   d) Median calculation method

9. The intersection point of the 'Less than Ogive' and 'More than Ogive' curves gives the:
   a) Mean
   b) Mode
   c) Median
   d) Range

10. If Σfᵢ = 50 and Σfᵢxᵢ = 1200, then the mean (x̄) calculated using the direct method is:
    a) 20
    b) 24
    c) 25
    d) 60

Answer Key for MCQs:

1. c) 20 ( (15+25)/2 = 40/2 = 20 )
2. d) Range (Range is a measure of dispersion/spread)
3. b) Frequency of the class preceding the modal class
4. b) Less than type
5. d) 28 ( Mode = 3 * Median - 2 * Mean = 3 * 26 - 2 * 25 = 78 - 50 = 28 )
6. c) 20-30 (Highest frequency is 15)
7. b) The class preceding the median class
8. b) Assumed Mean Method or Step-Deviation Method
9. c) Median
10. b) 24 ( Mean = Σfᵢxᵢ / Σfᵢ = 1200 / 50 = 24 )

Remember to practice solving problems using each method for Mean, Mode, and Median. Pay close attention to identifying the correct values (l, h, f, f₀, f₁, f₂, cf, N) for the formulas. Good luck with your preparation!