Class 10 Mathematics Notes Chapter 14 (Chapter 14) – Examplar Problems (English) Book

Alright class, let's dive deep into Chapter 14, Statistics, from your NCERT Exemplar. This is a crucial chapter, not just for your board exams but also frequently tested in various government entrance exams due to its practical applications in data analysis. We'll focus on the core concepts and problem-solving techniques highlighted in the Exemplar book.

Chapter 14: Statistics - Detailed Notes (Based on NCERT Exemplar)

Core Concepts:

This chapter primarily deals with the study of data. We focus on:

1. Measures of Central Tendency: Finding a single value that represents the center or typical value of a dataset. (Mean, Median, Mode)
2. Graphical Representation: Visualizing data using cumulative frequency curves (Ogives).

1. Mean (Average)

The mean provides a measure of the central location of the data.

• (a) Mean of Ungrouped Data:

  • Formula: x̄ = (Sum of all observations) / (Total number of observations) = Σxᵢ / n

• (b) Mean of Grouped Data: (When data is presented in frequency distribution tables)

  • i. Direct Method:

    • Used when the values of xᵢ (class marks) and fᵢ (frequencies) are small.
    • Class Mark (xᵢ): (Upper Class Limit + Lower Class Limit) / 2
    • Formula: x̄ = Σ(fᵢxᵢ) / Σfᵢ
    • Here, Σfᵢ is the total frequency (N).

  • ii. Assumed Mean Method (Shortcut Method):

    • Used when xᵢ and fᵢ values are large. Simplifies calculation.
    • Steps:
      1. Calculate class marks (xᵢ).
      2. Choose an 'Assumed Mean' (a), usually one of the middle class marks.
      3. Calculate deviations: dᵢ = xᵢ - a
      4. Calculate fᵢdᵢ.
    • Formula: x̄ = a + (Σ(fᵢdᵢ) / Σfᵢ)

  • iii. Step-Deviation Method:

    • Used when class sizes (h) are equal and dᵢ values are large multiples of h. Further simplifies calculation.
    • Steps:
      1. Follow steps 1-3 of the Assumed Mean method.
      2. Calculate uᵢ = dᵢ / h = (xᵢ - a) / h, where h is the class size (Upper Limit - Lower Limit).
      3. Calculate fᵢuᵢ.
    • Formula: x̄ = a + h * (Σ(fᵢuᵢ) / Σfᵢ)

  • Exemplar Focus: Problems might involve finding a missing frequency when the mean is given, or dealing with inclusive class intervals (convert them to exclusive first by subtracting 0.5 from lower limits and adding 0.5 to upper limits). Understand the properties: if each observation is increased/decreased/multiplied/divided by a constant k, the mean also changes similarly. The sum of deviations from the actual mean (Σfᵢ(xᵢ - x̄)) is always zero.

2. Mode

The mode is the value of the observation that occurs most frequently.

• (a) Mode of Ungrouped Data: Simply the observation with the highest frequency. A dataset can have more than one mode (multimodal) or no mode.

• (b) Mode of Grouped Data:

  • Modal Class: The class interval with the maximum frequency.
  • Formula: Mode = l + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] * h
    • l = Lower limit of the modal class
    • h = Class size (assuming equal class sizes)
    • f₁ = Frequency of the modal class
    • f₀ = Frequency of the class preceding the modal class
    • f₂ = Frequency of the class succeeding the modal class
  • Exemplar Focus: Correctly identifying l, f₁, f₀, f₂, and h is crucial. Be careful with classes at the beginning or end of the distribution (where f₀ or f₂ might be considered 0 if applicable, though usually the modal class isn't the very first or last).

3. Median

The median is the value of the middle-most observation when the data is arranged in ascending or descending order. It divides the data into two equal halves.

• (a) Median of Ungrouped Data:

  • Arrange data in ascending order.
  • If n (number of observations) is odd, Median = ((n+1)/2)-th observation.
  • If n is even, Median = Average of (n/2)-th and (n/2 + 1)-th observations.

• (b) Median of Grouped Data:

  • Steps:
    1. Prepare a cumulative frequency column (cf).
    2. Find the total frequency N = Σfᵢ.
    3. Calculate N/2.
    4. Identify the Median Class: The class whose cumulative frequency (cf) is just greater than or equal to N/2.
  • Formula: Median = l + [(N/2 - cf) / f] * h
    • l = Lower limit of the median class
    • N = Total frequency (Σfᵢ)
    • cf = Cumulative frequency of the class preceding the median class
    • f = Frequency of the median class
    • h = Class size (assuming equal class sizes)
  • Exemplar Focus: Calculating cumulative frequency correctly is key. Identifying the median class and the correct cf value (the one before the median class) are common points of error. Problems might involve finding missing frequencies when the median is given. The median is less affected by extreme values (outliers) compared to the mean.

4. Empirical Relationship Between Mean, Median, and Mode

For moderately skewed distributions, there's an approximate relationship:

• Mode ≈ 3 * Median - 2 * Mean
• This formula is useful for estimating one measure if the other two are known, or for checking the reasonableness of your calculated values. It's an empirical relationship, meaning it's based on observation, not a strict mathematical derivation, and holds best for unimodal, moderately skewed frequency distributions.

5. Graphical Representation: Cumulative Frequency Curves (Ogives)

An ogive is a graphical representation of a cumulative frequency distribution. It's typically S-shaped.

• (a) Less Than Type Ogive:

  • Plot points: (Upper Class Limit, Corresponding Cumulative Frequency)
  • Start the curve from the lower limit of the first class with a cumulative frequency of 0.
  • It's a rising curve.

• (b) More Than Type Ogive:

  • Calculate 'more than' cumulative frequencies (start from total frequency and subtract frequency of each class).
  • Plot points: (Lower Class Limit, Corresponding 'More Than' Cumulative Frequency)
  • End the curve at the upper limit of the last class with a cumulative frequency of 0.
  • It's a falling curve.

• Finding Median from Ogives:

  • The x-coordinate of the point of intersection of the 'Less Than' ogive and the 'More Than' ogive gives the Median.
  • Alternatively, locate N/2 on the vertical (cumulative frequency) axis on the 'Less Than' ogive. The corresponding x-coordinate on the horizontal axis is the Median.

• Exemplar Focus: Accurate plotting of points is essential. Understanding what the axes and points represent. Using the ogive to find the median or other partition values (like quartiles, though less common in Class 10).

Key Terms Recap:

• Class Interval: The range defining a class (e.g., 10-20).
• Class Size (h): Upper Limit - Lower Limit.
• Class Mark (xᵢ): Mid-point of a class interval.
• Frequency (fᵢ): Number of observations falling in a particular class.
• Cumulative Frequency (cf): Sum of frequencies up to a particular class (for 'less than' type).

Remember to practice problems from the Exemplar thoroughly, as they often test deeper understanding and application of these concepts. Pay attention to the wording of the questions.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed, similar to what you might find in competitive exams:

1. For the following distribution:

   Class	0-5	5-10	10-15	15-20	20-25
   Frequency	10	15	12	20	9
   The upper limit of the median class is:
   (a) 10
   (b) 15
   (c) 20
   (d) 25

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

3. While computing the mean of grouped data, we assume that the frequencies are:
   (a) Centred at the upper limits of the classes
   (b) Centred at the lower limits of the classes
   (c) Centred at the class marks of the classes
   (d) Evenly distributed throughout the class

4. The empirical relationship between the three measures of central tendency is:
   (a) 3 Median = Mode + 2 Mean
   (b) 2 Median = Mode + 3 Mean
   (c) Median = 3 Mode + 2 Mean
   (d) 3 Mode = Median + 2 Mean

5. The class mark of the class 15.5 - 20.5 is:
   (a) 15
   (b) 18
   (c) 20.5
   (d) 5

6. The intersection point of the 'Less than' ogive and 'More than' ogive for a given data set gives the:
   (a) Mean
   (b) Mode
   (c) Median
   (d) Range

7. If the mean of the data x₁, x₂, ..., xₙ is x̄, then the mean of ax₁, ax₂, ..., axₙ (where a is a constant) is:
   (a) x̄
   (b) ax̄
   (c) x̄ / a
   (d) x̄ + a

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

9. For plotting a 'Less than' Ogive, which of the following pairs is plotted?
   (a) (Lower Limit, Frequency)
   (b) (Upper Limit, Frequency)
   (c) (Lower Limit, Cumulative Frequency)
   (d) (Upper Limit, Cumulative Frequency)

10. If the mean and median of a distribution are 20 and 22 respectively, then the mode is approximately:
    (a) 24
    (b) 21
    (c) 26
    (d) 20.5

Answer Key for MCQs:

1. (b) 15 (N=66, N/2=33. cf: 10, 25, 37,... Median class is 10-15. Upper limit is 15)
2. (d) Range (Range is a measure of dispersion)
3. (c) Centred at the class marks of the classes
4. (a) 3 Median = Mode + 2 Mean
5. (b) 18 ( (15.5 + 20.5) / 2 = 36 / 2 = 18 )
6. (c) Median
7. (b) ax̄
8. (c) Frequency of the class preceding the modal class
9. (d) (Upper Limit, Cumulative Frequency)
10. (c) 26 (Mode ≈ 3 Median - 2 Mean = 3(22) - 2(20) = 66 - 40 = 26)

Study these notes carefully and practice the Exemplar problems. Let me know if any specific concept needs further clarification. Good luck with your preparation!