Question

Answer True or False for each of the following. If your answer is FALSE, provide an example that proves your point. If your answer is true, explain.

Given a typical set of numerical data with an ODD number of values.

1. The mean is always one of the data values.

2. The median is always one of the data values

3. The mode is always one of the data values.

4. The range measures the variability of the given data set.   

5. The mean is always the best measure of central tendency to use.

These are the answer choices for each question:

False. The range is a value representing the difference between the maximum and minimum value and tells us nothing else about the data set.           

False. If there is an outlier or extreme value then the mean will be pulled in the direction of the outlier and not be a good representation of data.

False. The mean is the average of all the values so it is not necessarily a data value.

False. There could be a situation where there is no mode.           

True. The range represents the difference between the largest and smallest value so it shows the spread of data, hence variability.  

False. The there could two values in the middle and you would have to find the average of the two values and therefore the median would not be a data value.

True. Since the mean is the average of the data values it is always the best value to use when representing the data.           

True. If the median is the value in the middle and you are dealing with an ODD number of data values then the median will be the data value in the middle.           

True. If there is a mode then it will be one of the data values since it is the data value that shows up the most.           

True. The mean is the average of the data values so it must be one of the values      

      [ Choose ]            False. The range is a value representing the difference between the maximum and minimum value and tells us nothing else about the data set.            False. If there is an outlier or extreme value then the mean will be pulled in the direction of the outlier and not be a good representation of data. Example 1b            False. The mean is the average of all the values so it is not necessarily a data value. For example 1a            False. There could be a situation where there is no mode.            True. The range represents the difference between the largest and smallest value so it shows the spread of data, hence variability.            False. The there could two values in the middle and you would have to find the average of the two values and therefore the median would not be a data value. Example 1c.            True. Since the mean is the average of the data values it is always the best value to use when representing the data.            True. If the median is the value in the middle and you are dealing with an ODD number of data values then the median will be the data value in the middle.            True. If there is a mode then it will be one of the data values since it is the data value that shows up the most.            True. The mean is the average of the data values so it must be one of the values      

Answer 1

To answer whether a statement is True or False, we solve each of the statements as follows

1. The mean is always one of the data values. False. The mean is the average of all the values so it is not necessarily a data value. For example, Take 4 values- 2,3,5,6. Now the mean of the 4 numbers is given by

Mean = (Total of all the given values)/(Number of values given)

Mean = (2 + 3 + 5 + 6)/4

Mean = 16/4 = 4

4 does not exist in the given data set, Hence the statement is false

2. The median is always one of the data values. False. The there could two values in the middle and you would have to find the average of the two values and therefore the median would not be a data value. For example, 2,3,5,6. The median is the middle value of the data set, As the number of values is even (n), We take the average of (n/2) and (n/2) + 1 values when placed in increasing order

Increasing Order of the data set- 2,3,5,6

Hence Average of 2nd and 3rd value = (3 + 5)/2 = 4

4 does not exist in the given data set, Hence the statement is false

3. The mode is always one of the data values. True. If there is a mode then it will be one of the data values since it is the data value that shows up the most

4. The range measures the variability of the given data set. True. The range represents the difference between the largest and smallest value so it shows the spread of data, hence variability

5. The mean is always the best measure of central tendency to use. True. Since the mean is the average of the data values it is always the best value to use when representing the data.