Median Class 11 Statistics Notes

1. Median:

• Median is the middle value in a set of data when the data points are arranged in ascending or descending order. It divides the data into two equal halves: 50% of the data lies below the median, and 50% lies above it.
• The median is less sensitive to extreme values (outliers) than the mean, making it a better measure of central tendency for skewed distributions.

2. Median for Ungrouped Data:

• For ungrouped data, the median is the value that separates the ordered data set into two equal halves.

Steps to find the median for ungrouped data:

1. Arrange the data in ascending or descending order.
2. If the number of observations (n) is odd, the median is the value at the position (n+1)/2(n+1)/2.
3. If the number of observations (n) is even, the median is the average of the values at the positions n/2n/2 and (n/2)+1(n/2) + 1.

Example:

• Data: 4, 7, 1, 9, 3
• Ordered data: 1, 3, 4, 7, 9
• Since n = 5 (odd), the median is at the 3rd position: 4.

For even data, e.g., 1, 3, 4, 7:

• n = 4 (even), the median is the average of the 2nd and 3rd values: 3+42=3.5\frac{3 + 4}{2} = 3.5.

3. Median for Grouped Data (Frequency Distribution):

• For grouped data, the median is determined using the cumulative frequency distribution.

Steps to find the median for grouped data:

1. Prepare the cumulative frequency table: Add a cumulative frequency column to the frequency table.
2. Find the median class:
   • The median class is the class whose cumulative frequency is greater than or equal to n/2n/2, where nn is the total number of observations.
3. Apply the median formula: Median=L+(n2−Ff)×h\text{Median} = L + \left( \frac{\frac{n}{2} – F}{f} \right) \times h where:
   • LL = Lower boundary of the median class
   • nn = Total number of observations
   • FF = Cumulative frequency of the class before the median class
   • ff = Frequency of the median class
   • hh = Class width (difference between upper and lower boundaries of the class)

Example:

• Total number of observations n=35n = 35.
• The median class is the one with a cumulative frequency greater than or equal to n/2=17.5n/2 = 17.5. So, the median class is 20 – 30.
• Using the formula: Median=20+(17.5−1312)×10=20+(4.512)×10=20+3.75=23.75\text{Median} = 20 + \left( \frac{17.5 – 13}{12} \right) \times 10 = 20 + \left( \frac{4.5}{12} \right) \times 10 = 20 + 3.75 = 23.75

Thus, the median is 23.75.

Mean Notes Class 11 Statistics

4. Properties of the Median:

• The median is not affected by extreme values (outliers).
• It divides the data into two equal halves.
• In a symmetric distribution, the median is the same as the mean.
• In a skewed distribution, the median is closer to the mode if the data is negatively skewed, or closer to the mean if it is positively skewed.
• For continuous data, the median represents the point where the total area under the frequency curve is divided into two equal parts.

5. Comparison with Mean and Mode:

• Mean: The arithmetic average, sensitive to extreme values.
• Mode: The value that occurs most frequently.
• Median: The middle value, dividing the data into two equal parts, and less influenced by outliers.

6. Uses of Median:

• The median is often used in skewed distributions or data with outliers.
• It helps determine the central tendency in real-world problems like income distribution, housing prices, etc.

7. Conclusion:

• The median is a robust measure of central tendency, especially for skewed or non-symmetric data distributions. It provides an accurate representation of the data’s center, especially when there are extreme values (outliers) that may distort the mean.

This was all about Median Class 11 Statistics Notes. If you have any doubts, you can either join my telegram channel or ask your doubts in the comments section.