Class 9 Mathematics Notes Chapter 14 (Statistics) – Mathematics Book

Alright class, let's focus on Chapter 14 - Statistics. This is a very practical chapter, not just for your Class 9 exams, but also a fundamental building block for many government exams where data interpretation and basic statistical concepts are frequently tested. Pay close attention to the definitions and methods.

Chapter 14: Statistics - Detailed Notes for Government Exam Preparation (Based on NCERT Class 9)

1. Introduction to Statistics

• Statistics: The branch of mathematics dealing with the collection, organization, analysis, interpretation, and presentation of data.
• Data: Facts or figures collected with a definite purpose. Data can be numerical or descriptive.
• Why Statistics? To understand patterns, make comparisons, draw conclusions, and make informed decisions based on data.

2. Collection of Data

• Primary Data: Data collected by the investigator himself/herself for a specific purpose. It's original data collected firsthand.
  • Example: Conducting a survey in your locality to find the number of families with two-wheelers.
• Secondary Data: Data collected by someone else (source) and used by the investigator. This data is already available.
  • Example: Using population data from the Census of India website for a project.
  • Caution: Secondary data should be used carefully, ensuring its reliability, suitability, and adequacy for the purpose.

3. Presentation of Data

Raw data collected is often large and difficult to understand. We need to organize and present it meaningfully.

• Raw Data: Data in its original form as collected.

• Array: Arranging raw numerical data in ascending or descending order.

• Range: The difference between the highest and lowest values (observations) in the data.

  • Range = Maximum Value - Minimum Value

• Frequency Distribution: A tabular arrangement of data showing the frequency (how many times) each observation or group of observations occurs.

  • Frequency: The number of times a particular observation occurs in the data.

  • Ungrouped Frequency Distribution Table: Used when the number of distinct observations is manageable. Lists each observation and its corresponding frequency. Tally marks are often used during construction.

    • Example Table Structure:

      Observation (x)	Tally Marks	Frequency (f)
      Value 1
      Value 2
      ...	...	...

  • Grouped Frequency Distribution Table: Used when the range of data is large. Data is grouped into 'classes' or 'class intervals'.

    • Class Interval: A range into which data values are grouped (e.g., 10-20, 20-30).
    • Class Limits: The lowest and highest values that can be included in a class interval.
      • Lower Class Limit: The smallest value in a class interval.
      • Upper Class Limit: The largest value in a class interval.
    • Class Size (or Class Width / Height): The difference between the true upper limit and the true lower limit of a class interval. For continuous classes like 10-20, 20-30, Class Size = Upper Limit - Lower Limit (e.g., 20 - 10 = 10).
    • Types of Class Intervals:
      • Exclusive Form (Continuous): The upper limit of one class is the lower limit of the next class (e.g., 10-20, 20-30, 30-40). An observation equal to the upper limit is included in the next class (e.g., 20 is included in 20-30, not 10-20). This is generally preferred for continuous data.
      • Inclusive Form (Discontinuous): There is a gap between the upper limit of one class and the lower limit of the next (e.g., 10-19, 20-29, 30-39). Both lower and upper limits are included in the class itself.
        • Conversion to Exclusive: If needed (e.g., for histograms), find the difference between the upper limit of one class and the lower limit of the next (e.g., 20 - 19 = 1). Half this difference (1/2 = 0.5) is subtracted from all lower limits and added to all upper limits (e.g., 10-19 becomes 9.5-19.5, 20-29 becomes 19.5-29.5).
    • Class Mark (or Mid-value / Mid-point): The central value of a class interval.
      • Class Mark = (Upper Class Limit + Lower Class Limit) / 2

4. Graphical Representation of Data

Visual representations make data easier to understand and compare.

• Bar Graph:
  • Used to represent categorical data (discrete data).
  • Consists of rectangular bars of uniform width.
  • The height (or length) of each bar is proportional to the value (frequency) it represents.
  • Bars are drawn with equal spacing between them on one axis (usually x-axis), and values are shown on the other axis (usually y-axis).
• Histogram:
  • Used to represent the frequency distribution of continuous grouped data (exclusive form).
  • Consists of rectangular bars adjacent to each other (no gaps).
  • The area of each bar is proportional to the frequency it represents.
  • Class intervals are marked on the horizontal axis, and corresponding frequencies on the vertical axis.
  • Important: If class intervals are of unequal width, the heights of the rectangles need to be adjusted proportionally (Frequency Density = Frequency / Class Width). The height represents frequency density in such cases. (This adjustment is less common in basic Class 9 questions but important conceptually).
  • Kink: If the first class interval does not start from zero, a kink (~) or break mark is shown on the horizontal axis near the origin.
• Frequency Polygon:
  • Another way to represent grouped frequency distribution (continuous).
  • Obtained by plotting the class marks against the corresponding frequencies and joining these points with straight line segments.
  • Can be drawn independently or by joining the mid-points of the tops of the rectangles in a histogram.
  • To complete the polygon, points corresponding to imaginary classes with zero frequency before the first class interval and after the last class interval are plotted and joined. (Class mark of preceding class, 0) and (Class mark of succeeding class, 0).
  • Useful for comparing the distributions of two or more datasets on the same graph.

5. Measures of Central Tendency

These are single values that attempt to describe the central position within a set of data.

• Mean (Arithmetic Mean or Average):
  • The sum of all observations divided by the total number of observations.
  • For ungrouped data: Mean (denoted by x̄) = (Sum of all observations) / (Total number of observations)
    • Formula: x̄ = Σxᵢ / n (where xᵢ represents individual observations, and n is the total number of observations)
  • For ungrouped frequency distribution: Mean (x̄) = (Sum of (frequency × observation)) / (Sum of frequencies)
    • Formula: x̄ = Σ(fᵢxᵢ) / Σfᵢ (where fᵢ is the frequency of the i-th observation xᵢ)
  • For grouped data: Mean (x̄) = (Sum of (frequency × class mark)) / (Sum of frequencies)
    • Formula: x̄ = Σ(fᵢxᵢ) / Σfᵢ (where fᵢ is the frequency of the i-th class, and xᵢ is the class mark of the i-th class)
• Median:
  • The value of the middle-most observation when the data is arranged in ascending or descending order.
  • For ungrouped data:
    1. Arrange the data in ascending or descending order.
    2. Let 'n' be the total number of observations.
    3. If 'n' is odd, Median = Value of the [(n+1)/2]th observation.
    4. If 'n' is even, Median = Average of the values of the (n/2)th and [(n/2) + 1]th observations.
       • Median = [Value of (n/2)th obs. + Value of ((n/2)+1)th obs.] / 2
• Mode:
  • The observation that occurs most frequently in the data.
  • A dataset can have one mode (unimodal), two modes (bimodal), or more than two modes (multimodal). It can also have no mode if all observations occur with the same frequency.
  • For ungrouped data: Simply identify the value(s) with the highest frequency.
  • For grouped data: The class interval with the highest frequency is called the modal class. (Calculating the exact mode within the modal class uses a formula usually taught in Class 10). For Class 9 level, identifying the modal class is usually sufficient.

Relevance for Government Exams:

• Direct questions on definitions (primary/secondary data, range, class mark, mean, median, mode).
• Calculating mean, median, mode for simple ungrouped data.
• Calculating mean for frequency distributions (ungrouped and grouped).
• Identifying class size, class limits, class marks from a grouped frequency table.
• Reading and interpreting Bar Graphs and Histograms (e.g., finding frequency for a class, comparing values).
• Understanding the difference between Bar Graphs and Histograms.
• Basic data interpretation questions often rely on these foundational concepts.

Practice MCQs

Here are 10 multiple-choice questions to test your understanding:

1. Data collected by the investigator directly for the first time for a specific purpose is called:
   a) Secondary Data
   b) Primary Data
   c) Raw Data
   d) Grouped Data

2. The difference between the maximum and minimum values of the observations in a dataset is called:
   a) Frequency
   b) Class Interval
   c) Range
   d) Mean

3. The class mark of the class interval 90-120 is:
   a) 90
   b) 105
   c) 115
   d) 120

4. In a frequency distribution, the mid-value (class mark) of a class is 10 and the width (class size) of the class is 6. The lower limit of the class is:
   a) 6
   b) 7
   c) 8
   d) 12

5. Which graphical representation is suitable for continuous grouped frequency distributions?
   a) Bar Graph
   b) Histogram
   c) Pie Chart
   d) Pictograph

6. Find the mean of the first five natural numbers (1, 2, 3, 4, 5).
   a) 2
   b) 2.5
   c) 3
   d) 3.5

7. The scores of a batsman in 5 matches are: 25, 10, 45, 50, 10. What is the mode of the scores?
   a) 50
   b) 25
   c) 45
   d) 10

8. What is the median of the data: 7, 10, 4, 3, 9, 11, 20?
   a) 4
   b) 9
   c) 7
   d) 10

9. In a histogram, the bars are drawn:
   a) With gaps between them
   b) With no gaps between them
   c) With variable width
   d) As line segments

10. The following marks were obtained by 10 students in a test: 8, 5, 9, 7, 2, 5, 8, 6, 5, 8. What is the frequency of the mark '8'?
    a) 2
    b) 3
    c) 4
    d) 1

Answer Key:

1. b) Primary Data
2. c) Range
3. b) 105 [(90+120)/2]
4. b) 7 [Let lower limit be L, upper limit be U. (L+U)/2 = 10 => L+U=20. U-L=6. Solving these, 2U=26 => U=13. L = U-6 = 13-6 = 7]
5. b) Histogram
6. c) 3 [(1+2+3+4+5)/5 = 15/5 = 3]
7. d) 10 [It occurs twice, more than any other score]
8. b) 9 [Arrange: 3, 4, 7, 9, 10, 11, 20. n=7 (odd). Median = (7+1)/2 = 4th term, which is 9]
9. b) With no gaps between them
10. b) 3 [The mark '8' appears 3 times]

Study these notes thoroughly. Practice constructing frequency tables and drawing the graphs. Focus on understanding the calculation methods for mean, median, and mode. Good luck with your preparation!