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Statistics - Collection and Presentation of Data

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Statistics is the branch of mathematics dealing with the collection, organization, analysis, interpretation, and presentation of data. Data can be categorized as Primary Data (collected directly by the investigator for a specific purpose) or Secondary Data (collected from an existing source like newspapers or the internet).

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Raw data is an unorganized collection of observations. To make it meaningful, it is arranged in ascending or descending order (an Array) or presented in a Frequency Distribution Table. Visually, this table uses tally marks where four vertical lines are crossed by a fifth diagonal line to represent a count of five, making it easy to read frequencies.

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Grouped Frequency Distribution is used for large data sets by grouping values into 'Class Intervals' such as 0−10,10−200-10, 10-20. The lower value is the Lower Class Limit and the higher value is the Upper Class Limit. In 'Exclusive' or continuous form, the upper limit of one class is the lower limit of the next; if a value equals the upper limit, it is visually placed in the next higher class interval.

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A Bar Graph is a visual representation consisting of vertical or horizontal rectangular bars of uniform width with equal spacing between them. The height of each bar represents the frequency of the specific category, making it easy to compare different discrete variables at a glance.

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A Histogram is a graphical representation of a continuous grouped frequency distribution. Unlike bar graphs, there are no gaps between the rectangular bars. The class intervals are plotted on the horizontal xx-axis and frequencies on the vertical yy-axis. Visually, if the first class doesn't start at zero, a 'kink' or 'zigzag' line is drawn on the xx-axis to show the break in the scale.

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A Frequency Polygon is a visual alternative to a histogram. It is formed by joining the mid-points (class marks) of the tops of the rectangles in a histogram with straight lines. To complete the polygon, it is extended to the xx-axis by joining it to the mid-points of imaginary classes with zero frequency at both ends, creating a closed geometric shape.

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Range is the simplest measure of dispersion, representing the spread of data. Visually, it is the distance on a number line from the smallest observation to the largest observation in the data set.

📐Formulae

Range=Maximum value−Minimum valueRange = Maximum\ value - Minimum\ value

Class Mark=Upper Class Limit+Lower Class Limit2Class\ Mark = \frac{Upper\ Class\ Limit + Lower\ Class\ Limit}{2}

Class Width=Upper Class Limit−Lower Class LimitClass\ Width = Upper\ Class\ Limit - Lower\ Class\ Limit

Adjusted Frequency (for Histogram with varying width)=Frequency of ClassWidth of Class×Minimum Class WidthAdjusted\ Frequency\ (for\ Histogram\ with\ varying\ width) = \frac{Frequency\ of\ Class}{Width\ of\ Class} \times Minimum\ Class\ Width

💡Examples

Problem 1:

The marks obtained by 10 students in a mathematics test are: 25,30,25,40,30,50,25,45,40,3025, 30, 25, 40, 30, 50, 25, 45, 40, 30. Construct a frequency distribution table and find the range.

Solution:

  1. Arrange the data in ascending order: 25,25,25,30,30,30,40,40,45,5025, 25, 25, 30, 30, 30, 40, 40, 45, 50. \n2. Count the frequency of each observation: \n- 2525: 33 times \n- 3030: 33 times \n- 4040: 22 times \n- 4545: 11 time \n- 5050: 11 time \n3. Create the table with columns 'Marks' and 'Frequency'. \n4. Calculate Range: \nRange=Maximum Mark−Minimum MarkRange = Maximum\ Mark - Minimum\ Mark \nRange=50−25=25Range = 50 - 25 = 25

Explanation:

To organize raw data, we first count how many times each value appears (frequency). The range shows the difference between the highest and lowest scores.

Problem 2:

In a grouped frequency distribution, the class intervals are 10−20,20−30,30−4010-20, 20-30, 30-40. Find the class mark for the interval 20−3020-30 and the class size.

Solution:

  1. To find the Class Mark (xix_i): \nxi=Lower Limit+Upper Limit2x_i = \frac{Lower\ Limit + Upper\ Limit}{2} \nxi=20+302=502=25x_i = \frac{20 + 30}{2} = \frac{50}{2} = 25 \n2. To find the Class Size (hh): \nh=Upper Limit−Lower Limith = Upper\ Limit - Lower\ Limit \nh=30−20=10h = 30 - 20 = 10

Explanation:

The class mark is the central value of a class interval, used for plotting frequency polygons. The class size is the uniform width of the intervals.