11.2 Apply Transformations to Data

ADDING A CONSTANT TO DATA VALUES

When a constant is added to every value in a data set, the following are true:

· The mean, median, and mode of the new data set can be obtained by _adding_ the same constant to the mean, median, and mode of the original data set.

· The range and standard deviation are _unchanged_.

Example 1

Add a constant to data values

Midterm Scores The data set gives the midterm scores for a chemistry class:

84, 89, 98, 86, 96, 83, 87, 93, 91, 93

Every student in the class has also completed a bonus assignment, so the teacher adds 5 points to each score. Find the mean, median, mode, range and standard deviation of the test scores with and without the bonus points.

Solution

	Scores without bonus	Scores with bonus
Mean	_90_	_90 + 5 = 95_
Median	_90_	_90 + 5 = 95_
Mode	_93_	_93 + 5 = 98_
Range	_15_	_15 (unchanged)_
Standard Deviation	_4.8_	_4.8 (unchanged)_

MULTIPLYING DATA VALUES BY A CONSTANT

When each value of a data set is multiplied by a constant, the new mean, median, mode, range, and standard deviation can be found by _multiplying_ each original statistic by the same constant.

Example 2

Multiply data values by a constant

Olympics The data set lists the winning distances (in meters) in the women's Olympic javelin throw from 1968 to 2004. Find the mean, median, mode, range, and standard deviation of the distances in meters and in yards. (Note: 1 m » 1.094 yd)

19.61, 21.03, 21.16, 22.41, 20.48, 22.24, 21.06, 20.56, 20.56, 19.59

Solution

	Distance in meters	Distance in yards
Mean	_20.87_	l.094(_20.87_) = _22.83_
Median	_20.80_	l.094(_20.80_) = _22.75_
Mode	_20.56_	l.094(_20.56_) = _22.49_
Range	_2.82_	l.094(_2.82_) = _3.09_
Standard Deviation	_0.89_	l.094(_0.89_) = _0.97_