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Statistics - Mode of Grouped Data

Grade 10CBSE

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

🔑Concepts

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Mode represents the value among the observations which occurs most frequently. In a frequency distribution of grouped data, it is not possible to determine the mode just by looking at the frequencies; we first identify the 'Modal Class' which is the class with the maximum frequency.

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The Modal Class is the specific interval that contains the mode. Visually, if you were to draw a histogram of the frequency distribution, the modal class corresponds to the tallest rectangular bar in the graph.

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Lower Limit (ll): This is the boundary value that starts the modal class. On a horizontal x-axis of a distribution graph, ll is the point marking the beginning of the highest frequency interval.

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Frequency Identification: Three specific frequencies are required for the calculation. f1f_1 is the frequency of the modal class (the highest value), f0f_0 is the frequency of the class immediately preceding the modal class, and f2f_2 is the frequency of the class immediately following the modal class.

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Class Size (hh): This represents the width of the class intervals. It is visually the width of the bars in a histogram. For the formula to be applied correctly, it is assumed that the class sizes are uniform throughout the data set.

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Continuous Classes: The mode formula requires classes to be continuous (e.g., 0-10, 10-20). If classes are discontinuous (e.g., 1-10, 11-20), they must be converted by subtracting 0.50.5 from the lower limits and adding 0.50.5 to the upper limits to create a seamless boundary.

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Empirical Relationship: There is a theoretical relationship between the three measures of central tendency (Mean, Median, and Mode) expressed as 3×Median=Mode+2×Mean3 \times Median = Mode + 2 \times Mean. This is used to estimate one value when the other two are known.

📐Formulae

Mode=l+(f1−f02f1−f0−f2)×hMode = l + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h

3×Median=Mode+2×Mean3 \times Median = Mode + 2 \times Mean

h=Upper Limit−Lower Limith = \text{Upper Limit} - \text{Lower Limit}

💡Examples

Problem 1:

The following data gives the information on the observed lifetimes (in hours) of 225 electrical components:

  • 0-20: 10
  • 20-40: 35
  • 40-60: 52
  • 60-80: 61
  • 80-100: 38
  • 100-120: 29 Determine the modal lifetimes of the components.

Solution:

  1. Identify the maximum frequency: The maximum frequency is 6161, which belongs to the class interval 60−8060-80.
  2. Therefore, the Modal Class is 60−8060-80.
  3. Identify the values:
  • Lower limit of modal class (ll) = 6060
  • Frequency of modal class (f1f_1) = 6161
  • Frequency of preceding class (f0f_0) = 5252
  • Frequency of succeeding class (f2f_2) = 3838
  • Class size (hh) = 2020
  1. Apply the formula: Mode=60+(61−522(61)−52−38)×20Mode = 60 + \left( \frac{61 - 52}{2(61) - 52 - 38} \right) \times 20 Mode=60+(9122−90)×20Mode = 60 + \left( \frac{9}{122 - 90} \right) \times 20 Mode=60+(932)×20Mode = 60 + \left( \frac{9}{32} \right) \times 20 Mode=60+18032Mode = 60 + \frac{180}{32} Mode=60+5.625=65.625Mode = 60 + 5.625 = 65.625 Hence, the modal lifetime of the components is 65.62565.625 hours.

Explanation:

To find the mode, we first locate the interval with the highest frequency (the modal class). Then, we substitute the lower limit, the class size, and the frequencies of the modal, preceding, and succeeding classes into the standard mode formula for grouped data.

Problem 2:

In a distribution, the Mean is 2424 and the Median is 2626. Using the empirical relationship, find the Mode.

Solution:

  1. Given values: Mean=24Mean = 24, Median=26Median = 26.
  2. Use the empirical formula: 3×Median=Mode+2×Mean3 \times Median = Mode + 2 \times Mean
  3. Substitute the values: 3×26=Mode+2×243 \times 26 = Mode + 2 \times 24 78=Mode+4878 = Mode + 48
  4. Solve for Mode: Mode=78−48Mode = 78 - 48 Mode=30Mode = 30 Therefore, the Mode of the distribution is 3030.

Explanation:

This example utilizes the relationship between the three measures of central tendency. By substituting the known Mean and Median into the empirical equation, we can algebraically solve for the Mode.