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Question

Consider the following observations regarding the median of a data set:

__Observation I__: The median is defined only for numerical data.

__Observation II__: For an odd number of data points, the median is the central value after rearranging, whereas for an even number of data points, the median is the mean of two central values after rearranging.

__Observation III__: Unlike mode, the median may or may not be the same as an existing data point.

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Solution

The correct option is **B** All the observations are correct.

Here, the central values are 6 and 8.

Median = Mean of two central values

⇒ Median =6+82

=142

=7

→ 7 is not present in the given data set.

Consider another arranged data set in which two central values are same.

Here, the central values are 5 and 5.

∴ Median =5+52

=102

=5

→ Observe that in this case the median of even numbered data set is same as one of the data points.

Hence, median may or may not be same as an existing data point.

∴ All the observations are correct.

Let's go through the mentioned observations one by one.

__Observation I__ :

Determination of median involves arranging the data in an ascending/descending order. Hence, it is only possible for numerical data.

__Observation II__ :

For an odd number of data, a single middle value exists after rearranging the data set. Whereas, for an even number of data set two middle values are found after rearranging.

By definition, for an even-numbered data set, the median is the mean/average of two central values of the rearranged data.

__Observation III__ :

For odd-numbered data, the median is the middle value of the rearranged data.

Hence, in this case, the median is the same as one of the data points.

For even number data, as the median is the average of two central values, it may or may not be the same as any of the data points.

For example, consider the following arranged data points:

5 | 6 | 8 | 10 |

Here, the central values are 6 and 8.

Median = Mean of two central values

⇒ Median =6+82

=142

=7

→ 7 is not present in the given data set.

Consider another arranged data set in which two central values are same.

3 | 5 | 5 | 8 |

Here, the central values are 5 and 5.

∴ Median =5+52

=102

=5

→ Observe that in this case the median of even numbered data set is same as one of the data points.

Hence, median may or may not be same as an existing data point.

∴ All the observations are correct.

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