Statistics and Probability Class 9 NCERT — Complete Guide

Data Presentation and Representation

In statistics, raw data must be formatted to derive meaningful conclusions. CBSE Class 9 emphasizes representing data in graphical formats. The primary methods include Bar Graphs, Histograms, and Frequency Polygons. A Bar Graph is ideal for displaying qualitative or discrete categories of data using uniformly spaced bars. On the other hand, a Histogram is designed for grouped continuous data, where class intervals are plotted on the horizontal axis (X-axis) and frequency on the vertical axis (Y-axis). Because the intervals are continuous, there are no gaps between consecutive bars. A Frequency Polygon is a line graph constructed by connecting the midpoints (class marks) of the upper bounds of a histogram, making it easier to analyze general data distribution trends.

Essential Concepts and Formulas

Class Mark

The midpoint of a class interval, calculated using the formula: Class Mark = (Upper Limit + Lower Limit) / 2.

Class Width (or Class Size)

The difference between the upper limit and the lower limit of a given class interval.

Empirical Probability

The probability of an event based on actual experimental trials, calculated as: P(E) = (Number of trials in which event E occurred) / (Total number of trials).

Range

The difference between the highest (maximum) value and the lowest (minimum) value in a dataset.

Drawing Histograms with Varying Class Widths

1. Identify the Minimum Class Size — Examine the class intervals in your grouped data table. Find the smallest class interval width (let this value be 'h').
2. Calculate Adjusted Frequencies — For each class interval, calculate its adjusted frequency to normalize the height using the formula: Adjusted Frequency = (Original Frequency / Width of Interval) * Minimum Class Width.
3. Plot Class Intervals on the X-axis — Mark the class boundaries clearly on the horizontal axis. Draw a broken line (kink) near the origin if the values do not start from zero.
4. Draw the Normalized Bars — Draw rectangular columns where the width corresponds to the class interval and the height represents the calculated adjusted frequency.

Pro-Tips to Avoid Common Mistakes in Board Exams

1. Draw the Kink (Broken Line): If your class intervals start from a high number (e.g., 100-110) rather than 0, always draw a kink symbol (a squiggly break) on the X-axis to show that the scale starts away from the origin.
2. Check Probability Limits: Remember that the probability of any event must strictly lie between 0 and 1 ($0 \le P(E) \le 1$). If your answer is negative or greater than 1, you made a calculation error!
3. Sum of Probabilities: The sum of probabilities of all possible elementary events in an experiment is always equal to 1. Use this rule to verify your calculations.

Practice Questions with Solutions

• Q: A coin is tossed 400 times with the following frequencies: Head: 185, Tail: 215. What is the empirical probability of getting a Tail?
  A: Step 1: Write down the total number of experimental trials. Total trials = 400.
  Step 2: Identify the frequency of the favorable event (getting a Tail). Frequency of Tail = 215.
  Step 3: Apply the empirical probability formula: P(Tail) = Frequency of Tail / Total trials.
  Step 4: Calculate: P(Tail) = 215 / 400 = 43 / 80 = 0.5375.
  Final answer: The empirical probability of getting a Tail is 0.5375.

• Q: Determine the class mark for the class interval 135 - 145.
  A: Step 1: Identify the lower limit and upper limit. Lower limit = 135, Upper limit = 145.
  Step 2: Use the class mark formula: Class Mark = (Lower Limit + Upper Limit) / 2.
  Step 3: Calculate the value: Class Mark = (135 + 145) / 2 = 280 / 2 = 140.
  Final answer: The class mark of the interval 135 - 145 is 140.

• Q: In a survey of 150 students, 90 students preferred studying online while 60 preferred physical classes. Find the probability that a student selected at random prefers physical classes.
  A: Step 1: Identify total outcomes. Total students surveyed = 150.
  Step 2: Identify favorable outcomes. Students preferring physical classes = 60.
  Step 3: Use the empirical probability formula: P(Physical) = Favorable outcomes / Total outcomes = 60 / 150.
  Step 4: Simplify the fraction: 60 / 150 = 6 / 15 = 2 / 5 = 0.4.
  Final answer: The probability that a randomly chosen student prefers physical classes is 0.4.

• Q: Find the range of the following dataset: 12, 25, 8, 47, 33, 19, 52, 21.
  A: Step 1: Identify the maximum (highest) value in the dataset. Max value = 52.
  Step 2: Identify the minimum (lowest) value in the dataset. Min value = 8.
  Step 3: Apply the range formula: Range = Maximum value - Minimum value.
  Step 4: Calculate: Range = 52 - 8 = 44.
  Final answer: The range of the given dataset is 44.

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