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Statistics - Graphical Representation (Histograms, Frequency Polygons, Ogives)

Grade 10ICSE

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

🔑Concepts

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Continuous Classes and Adjustment Factor: For graphical representation like Histograms, data must be in a continuous format. If classes are discontinuous (e.g., 10−19,20−2910-19, 20-29), an adjustment factor is calculated as half the difference between the lower limit of a class and the upper limit of the previous class. Visually, this closes the 'gaps' between bars in a histogram by extending the boundaries by 0.50.5 units (or the calculated factor) in both directions.

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Histogram for Equal and Unequal Class Intervals: A Histogram consists of a set of adjacent rectangles where the base represents the class interval on the x-axis and the height represents the frequency on the y-axis. If class intervals are unequal, the heights of the rectangles must be adjusted using the 'Frequency Density' principle, where the height of a rectangle is proportional to the ratio of the frequency to the class width. Visually, a 'kink' or zig-zag line is drawn on the x-axis if the scale does not start from zero.

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Finding Mode Graphically from a Histogram: The mode of a distribution can be estimated by identifying the modal class, which is the tallest rectangle in the histogram. By drawing two straight diagonal lines from the top corners of the tallest rectangle to the opposite top corners of the two immediate neighboring rectangles, the x-coordinate of the intersection point of these diagonal lines gives the Mode. This visual method represents the point of highest frequency density.

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Frequency Polygon: This is a line graph formed by joining the mid-points (class marks) of the tops of the rectangles in a histogram with straight lines. To complete the polygon, the ends are extended to join the x-axis at the mid-points of imaginary classes preceding the first and succeeding the last given classes. Visually, it provides a 'shape' to the distribution and is useful for comparing two different datasets on the same axes.

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Cumulative Frequency and the Ogive: An Ogive is a smooth, freehand 'S-shaped' curve obtained by plotting the upper limits of class intervals on the x-axis and their corresponding 'less than' cumulative frequencies on the y-axis. To start the curve on the x-axis, the lower limit of the first class is plotted with a cumulative frequency of zero. This curve visually represents how frequencies accumulate across the range of data.

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Median and Quartiles from an Ogive: The Ogive is used to locate measures of central tendency and partition values. To find the Median, locate the value N2\frac{N}{2} on the y-axis (where NN is total frequency), draw a horizontal line to meet the curve, and then drop a vertical line to the x-axis. The value on the x-axis is the Median. Similarly, the Lower Quartile (Q1Q_1) is found at N4\frac{N}{4} and the Upper Quartile (Q3Q_3) is found at 3N4\frac{3N}{4} on the y-axis.

📐Formulae

Class Mark(xi)=Upper Limit+Lower Limit2\text{Class Mark} (x_i) = \frac{\text{Upper Limit} + \text{Lower Limit}}{2}

Adjustment Factor=Lower Limit of a class−Upper Limit of previous class2\text{Adjustment Factor} = \frac{\text{Lower Limit of a class} - \text{Upper Limit of previous class}}{2}

Adjusted Frequency (for unequal classes)=Frequency of the classWidth of the class×Minimum class width\text{Adjusted Frequency (for unequal classes)} = \frac{\text{Frequency of the class}}{\text{Width of the class}} \times \text{Minimum class width}

Total Frequency(N)=∑fi\text{Total Frequency} (N) = \sum f_i

Median Position=(N2)th term\text{Median Position} = \left(\frac{N}{2}\right)^{th} \text{ term}

Lower Quartile (Q1) Position=(N4)th term\text{Lower Quartile (Q1) Position} = \left(\frac{N}{4}\right)^{th} \text{ term}

Upper Quartile (Q3) Position=(3N4)th term\text{Upper Quartile (Q3) Position} = \left(\frac{3N}{4}\right)^{th} \text{ term}

Inter-quartile Range=Q3−Q1\text{Inter-quartile Range} = Q_3 - Q_1

💡Examples

Problem 1:

Draw a histogram for the following distribution and find the Mode graphically:

Class0-1010-2020-3030-4040-50
Freq51220158

Solution:

  1. Plot the class intervals 0−10,10−20,…,40−500-10, 10-20, \dots, 40-50 on the x-axis.
  2. Plot frequencies on the y-axis (Scale: 1 cm=5 units1 \text{ cm} = 5 \text{ units}).
  3. Construct rectangles for each class with heights 5,12,20,15,5, 12, 20, 15, and 88.
  4. Identify the modal class: 20−3020-30 (tallest rectangle, height =20= 20).
  5. Draw a line from the top-left corner of the 20−3020-30 rectangle to the top-left corner of the 30−4030-40 rectangle.
  6. Draw another line from the top-right corner of the 20−3020-30 rectangle to the top-right corner of the 10−2010-20 rectangle.
  7. Locate the intersection point of these two diagonal lines.
  8. Draw a perpendicular line from this intersection to the x-axis. The value on the x-axis is approximately 2626.

Mode≈26\text{Mode} \approx 26.

Explanation:

The mode is found by examining the area of highest frequency. The intersection of the diagonal lines accounts for the influence of the frequencies of classes immediately preceding and following the modal class.

Problem 2:

Given the following frequency distribution, draw an Ogive and estimate the Median:

Marks10-2020-3030-4040-5050-60
Students4915102

Solution:

  1. Calculate Cumulative Frequencies (CFCF):
    • 10−20:410-20: 4
    • 20−30:4+9=1320-30: 4 + 9 = 13
    • 30−40:13+15=2830-40: 13 + 15 = 28
    • 40−50:28+10=3840-50: 28 + 10 = 38
    • 50−60:38+2=4050-60: 38 + 2 = 40
  2. Points to plot (UpperLimit,CF)(Upper Limit, CF): (20,4),(30,13),(40,28),(50,38),(60,40)(20, 4), (30, 13), (40, 28), (50, 38), (60, 40).
  3. Include the starting point: (10,0)(10, 0).
  4. Plot these points on a graph and join them with a smooth freehand curve.
  5. Total frequency N=40N = 40. Median position =N2=402=20th= \frac{N}{2} = \frac{40}{2} = 20^{th} term.
  6. On the y-axis, find the value 2020. Draw a horizontal line to the Ogive.
  7. From the intersection point on the Ogive, drop a vertical line to the x-axis. The value on the x-axis is the Median marks.

Median≈34.6\text{Median} \approx 34.6

Explanation:

The Ogive represents the cumulative distribution. By finding the middle point of the total frequency (N/2N/2) on the y-axis, we can trace back to the x-axis to find the value below which 50%50\% of the data lies.