Effect of linear transformations on sample statistics

Occasionally, statistics will be collected on variables measured in one unit, and needs to be converted to another unit. For example, lengths in inches need to be converted to lengths in centimeters.

What is the effect of adding the same value to all measurements? Since all values are shifted the same amount, the measures of location (mean, median, mode, percentiles ) will all shift the same amount. The spread will not change, so the std deviation will not change.

What is the effect of multiplying all measurements by the same value? This will shift all values by the same multiplicative values, and so the same effect is felt by the measures of location (mean, median, mode, percentiles). Multiplying spreads out the values more, so the std deviation is also multiplied.

In general, if the transform is $X_{new}= a + b X_{old}$ where $a$ and $b$ are known values, then

• $\overline{X}_{new }= a + b \times \overline{X}_{old}$

• $median_{new}= a + b \times median_{old}$

• $mode_{new}= a + b \times mode_{old}$

• $percentile_{new}= a + b percentile_{old}$

• $s_{new}= b s_{old}$
  • Note that the additive change does not affect the std deviation.
  • If $b<0$ , then the standard deviation is multiplied by the absolute value of $b$ .

Here are some examples

• The average birth weight of male babies is around 6 lbs and 4 oz with a standard deviation of 4 oz. What is the mean and standard deviation in kg? [Remember that 1 kg = 2.2 lbs, and there are 16 oz in a 1 lb.

• The average temperature in Winnipeg in October is 36 degrees F with a standard deviation of 2 degrees F. What is the average temperature and standard deviation in degrees C. [Recall that degrees F = degrees C*9/5 + 32.]

Important The above ONLY works for linear transformations - it will NOT WORK for non-linear transformations. For example, find the mean and standard deviation of the numbers 1, 2, 3, 4, 5 and the numbers 1, 4, 9, 16, 25. The latter are the square of the former. Now compute the square of the mean and the standard deviation from the first set - the won't be equal to the mean and standard deviation from the second set.