a)
To find: A set of data where the standard deviation is less than the interquartile range of the data distribution.
The set of data where the standard deviation is less than the interquartile range is shown below
1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7.
Given:
The collected set of data is 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7.
Concept used:
The standard deviation and interquartile range are measures of variability, the standard deviation is nothing but the average deviation of all values with respect to the mean and the interquartile range is the range of the middle 50% values of the data distribution.
The standard deviation is calculated using the formula
The Interquartile range is the difference between the third Quartile and the first Quartile, where the first quartile divides the data distribution bottom 25% and top 75% of the data and the quartile three divides the data bottom 75% and top 25% of the data.
Calculation:
The mean of the set of data is calculated as shown below
The standard deviation of the set of data is calculated as shown below
The standard deviation of the data set is 1.959.
There are a total of 15 data points given in the data distribution arranging them in ascending order, the median of the data distribution is the middle value which is at the 8th position, which is 5.
The first quartile divides the bottom 25% of the data and the top 75% of the data, it is nothing but the middle value of the values of the first half of the data i.e.
1, 2, 3, 3, 4, 4, 5, 5. There are a total of 8 data values and the middle value is the average of data values at 4th and 5th positions.
The first quartile is 3.5.
The third quartile divides the bottom 75% of the data and the top 25% of the data, it is nothing but the middle value of the values of the second half of the data i.e.
5, 6, 6, 6, 7, 7, 7, 7. There are a total of 8 data values and the middle value is the average of data values at 4th and 5th positions.
The third quartile is 6.5.
The Interquartile range of the set of data is the difference between the third quartile and first quartile and is calculated as shown below
Conclusion:
The collected data set 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7 satisfy the condition that the interquartile range is more than the standard deviation.
b)
To find: A set of data where the interquartile range is less than the standard deviation of the data distribution.
The set of data where the interquartile range is less than the standard deviation is shown below
1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 10, 15.
Given:
The collected set of data is 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 10, 15.
Concept used:
The standard deviation and interquartile range are measures of variability, the standard deviation is nothing but the average deviation of all values with respect to the mean and the interquartile range is the range of the middle 50% values of the data distribution.
The standard deviation is calculated using the formula
The Interquartile range is the difference between the third Quartile and the first Quartile, where the first quartile divides the data distribution bottom 25% and top 75% of the data and the quartile three divides the data bottom 75% and top 25% of the data.
Calculation:
The mean of the set of data is calculated as shown below
The standard deviation of the set of data is calculated as shown below
The standard deviation of the data set is 3.439.
There are a total of 15 data points given in the data distribution arranging them in ascending order, the median of the data distribution is the middle value which is at the 8th position, which is 5.
The first quartile divides the bottom 25% of the data and the top 75% of the data, it is nothing but the middle value of the values of the first half of the data i.e.
1, 2, 3, 3, 4, 4, 5, 5. There are a total of 8 data values and the middle value is the average of data values at 4th and 5th positions.
The first quartile is 3.5.
The third quartile divides the bottom 75% of the data and the top 25% of the data, it is nothing but the middle value of the values of the second half of the data i.e.
5, 6, 6, 6, 7, 7, 10, 15. There are a total of 8 data values and the middle value is the average of data values at 4th and 5th positions.
The third quartile is 6.5.
The Interquartile range of the set of data is the difference between the third quartile and first quartile and is calculated as shown below
Conclusion:
The collected data set 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 10, 15 satisfy the condition that the standard deviation is more than the interquartile range.
Chapter 10 Solutions
PRECALCULUS:GRAPHICAL,...-NASTA ED.
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