13.1-Mean of Grouped Data

13.1-Mean of Grouped Data Important Formulae

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Grade 10 → Math → Statistics → 13.1-Mean of Grouped Data

After successful completion of this topic, you should be able to:

  • Apply direct method in order to calculate the mean of the grouped data.
  • Apply assumed mean method in order to calculate the mean for a grouped data.

The mean of grouped data is a statistical measure representing the average of a dataset organized into classes or groups. This approach is particularly useful when dealing with large datasets where individual values may not provide significant insights compared to the overall trends.

To calculate the mean of grouped data, we follow these steps:

  1. Identify the classes: Organize the data into intervals (classes) based on the range of values.
  2. Determine the midpoints: For each class, calculate the midpoint (class mark). The midpoint is given by the formula:

Midpoint $x_i = \frac{{\text{Lower limit} + \text{Upper limit}}}{2}$

  1. Frequency count: Count the number of data points (frequency) in each class.
  2. Create a table: Construct a frequency distribution table that includes the class intervals, midpoints, and their corresponding frequencies.
Class Interval Midpoint $x_i$ Frequency $f_i$ Product $f_i \cdot x_i$
10 - 20 15 5 75
20 - 30 25 10 250
30 - 40 35 8 280

In this example, the product of frequency and midpoint for each class is calculated. To find the mean, we need to sum these products and the frequencies.

  1. Calculate the total: Find the sum of all frequencies $N = \Sigma f_i$ and the sum of the products $ \Sigma (f_i \cdot x_i) $.

Total Frequency $N = \Sigma f_i = f_1 + f_2 + f_3 = 5 + 10 + 8 = 23$

Total $ \Sigma (f_i \cdot x_i) = 75 + 250 + 280 = 605$

  1. Apply the mean formula: The mean $\bar{x}$ of the grouped data can be calculated using the formula:

Mean $\bar{x} = \frac{{\Sigma (f_i \cdot x_i)}}{{\Sigma f_i}} = \frac{605}{23} \approx 26.35$

This result represents the mean value of the grouped data set. It is important to note that the mean calculated in this manner provides a good estimate of the average value, especially in large datasets.


Geometric visualisation of the mode, median and mean of an arbitrary probability density function.
Cmglee, CC BY-SA 3.0, via Wikimedia Commons

A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
Number of plants Number of houses
0-2      1
2-4      2
4-6      1
6-8      5
8-10      6
10-12      2
12-14      3
Which method did you use for finding the mean, and why?

Solution:

Data Collection for Number of Plants

Class intervals and frequencies:

  • 0-2 plants: 1 house
  • 2-4 plants: 2 houses
  • 4-6 plants: 1 house
  • 6-8 plants: 5 houses
  • 8-10 plants: 6 houses
  • 10-12 plants: 2 houses
  • 12-14 plants: 3 houses
Calculating Midpoints

Midpoint for each class interval:

  • 0-2: (0+2)/2 = 1
  • 2-4: (2+4)/2 = 3
  • 4-6: (4+6)/2 = 5
  • 6-8: (6+8)/2 = 7
  • 8-10: (8+10)/2 = 9
  • 10-12: (10+12)/2 = 11
  • 12-14: (12+14)/2 = 13
Finding the Mean

Using the formula for mean:

Mean = (Σ(f × x)) / N

Where f = frequency and x = midpoint, N = total number of houses = 20

Calculation Steps
  • Calculate Σ(f × x) for all intervals.
  • Divide by total number of houses (N).

Suggest improvements/Report errors

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