Statistics | Class 11 Mathematics Notes

Mean Deviation for Grouped Data

Mean deviation for grouped data is calculated by considering the class mid-points as representative values of the classes. The process involves first finding the mean or median of the grouped data, then calculating the absolute deviations of class mid-points from this central value, multiplying by the class frequencies, summing these products, and finally dividing by the total number of observations. The chapter provides detailed step-by-step procedures and examples to illustrate this method. For example, given frequency distributions of marks or incomes, students learn to compute the mean deviation about the mean or median using tabular methods. The chapter also explains the use of cumulative frequencies to find the median class in grouped data. This approach helps in understanding the variability in data that is presented in intervals rather than individual values, which is common in real-world data collection.

📊 Diagram: Table 1 on page 8; Table 2 on page 8; Figure 8 on page 5

🧪 Activity: Students organize collected data into class intervals and compute mean deviation to analyze variability.

🔗 Connection: This section prepares students for understanding variance and standard deviation calculations for grouped data.

Table on page 8 (7×15)

| | x i | | f i | | | fx i i | | | x −x i | | | f x −x i i | | |

| 2 2 4 5.5 11 5 8 40 2.5 20 6 10 60 1.5 15 8 7 56 0.5 3.5 10 8 80 2.5 20 12 5 60 4.5 22.5 40 300 92 6 6 6 N = ∑ f =40 ∑ f x =300 ∑ f x −x =92 , , i i i i i i=1 i=1 i=1 1 6 1 Therefore x = ∑ f ix i = ×300=7.5 N 40 i=1 1 6 1 and M. D. ( x) = ∑ f x −x = × 92 = 2.3 i i N 40 i=1 Example 5 Find the mean deviation about the median for the following data: x 3 6 9 12 13 15 21 22 i f 3 4 5 2 4 5 4 3 i Solution The given observations are already in ascending order. Adding a row corresponding to cumulative frequencies to the given data, we get (Table 13.2). Table 13.2 x 3 6 9 12 13 15 21 22 i f 3 4 5 2 4 5 4 3 i | 2 5 6 8 10 12 | | 2 8 10 7 8 5 | | | 4 40 60 56 80 60 | | | 5.5 2.5 1.5 0.5 2.5 4.5 | | | 11 20 15 3.5 20 22.5 | | |

| | | x i | | 3 | 6 | | 9 | 12 | | 13 | 15 | 21 | 22 | | | | | f i | | 3 | 4 | | 5 | 2 | | 4 | 5 | 4 | 3 | |

Table on page 8 (1×9)

| x i | 3 | 6 | 9 | 12 | 13 | 15 | 21 | 22 |

| f i | 3 | 4 | 5 | 2 | 4 | 5 | 4 | 3 |