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Changes in mean

The mean has many characteristics that are important. In general, these characteristics result from the fact that every score in the distribution contributes to the value of the mean. Specifically, every score adds to the total $(\Sigma\text{X})$ and every score contributes one point to the number of scores (n). These two values ($\Sigma\text{X}$ and n) determine the value of the mean. We now discuss four of the more important characteristics of the mean:

• Changing a score.
• Introducing a new score or removing a score.
• Adding or subtracting a constant from each score.
• Multiplying or dividing each score by a constant.

CHANGING A SCORE
Changing the value of any score will change the mean.

A sample of quiz scores for a psychology lab section consists of 9, 8, 7, 5, and 1. Note that the sample consists of n=5 scores with $\Sigma\text{X}=30$. The mean for this example is

$M=\frac{\Sigma\text{X}}{n}=\frac{30}{5}=6$

Now suppose that the score of X=1 is changed to X=8. Note that we have added 7 points to this individual's score, which will also add 7 points to the total $(\Sigma\text{X})$. After changing the score, the new distribution consists of:

9, 8, 7, 5, 8

There are still n=5 scores, but now the total is $\Sigma\text{X}=37$. Thus, the new mean is

$M=\frac{\Sigma\text{X}}{n}=\frac{37}{5}=7.40$

Notice that changing a single score in the sample has produced a new mean. You should recognize that changing any score also changes the value of $\Sigma\text{X}$ (the sum of the scores), and thus always changes the value of the mean.

INTRODUCING A NEW SCORE OR REMOVING A SCORE
In general, the mean is determined by two values: $\Sigma\text{X}$ and n. Whenever either of these values is changed, the mean also is changed. If you add a new score (or take away a score), you will change both $\Sigma\text{X}$ and n, and you must compute the new mean using the changed values.

A sample of quiz scores for a psychology lab section consists of 3, 6, 6, 9, and 11. Note that the sample consists of n=5 scores with $\Sigma\text{X}=35$. The mean for this example is

$M=\frac{\Sigma\text{X}}{n}=\frac{35}{5}=7$

What will happen to the mean if a new score of X=13 is added to the sample?

To find the new sample mean, we must determine how the values for n and $\Sigma\text{X}$ will be changed by a new score. We begin with the original sample and then consider the effect of adding the new score. The original sample had n=5 scores, so adding one new score will produce n=6. Similarly, the original sample had $\Sigma\text{X}=35$. Adding a score of X=13 will produce a new sum of $\Sigma\text{X}=35+13=48$.

Finally, the new mean is computed using the new values for $\Sigma\text{X}$ and n.

$M=\frac{\Sigma\text{X}}{n}=\frac{48}{6}=8$

ADDING OR SUBTRACTING A CONSTANT FROM EACH SCORE.
If a constant value is added to every score in a distribution, the same constant will be added to the mean. similarly, if you subtract a constant from every score, the same constant will be subtracted from the mean.

We are interested on discussing a research study in which participants consistently have better memory for humorous sentences than for nonhumorous sentences. The following table shows the result for a sample of n=6 participants. The first column shows their memory scores for nonhumorous sentences:

 Number of sentences recalled for humorous and nonhumorous sentences Participant Nonhumorous sentences Humorous sentences A 4 6 B 2 4 C 3 5 D 3 5 E 2 4 F 3 5

Note that the total number of sentences recalled is $\Sigma\text{X}=17$ for a sample of n=6 participants, so the mean is $M=\frac{\Sigma\text{X}}{n}=\frac{17}{6}=2.83$. Now suppose that the effect of humor is to add a constant amount (2 points) to each individual's memory score. The resulting scores for humorous sentences are shown in the second column of the table.

For these scores, the 6 participants recalled a total of $\Sigma\text{X}=29$ sentences, so the mean is $M=\frac{\Sigma\text{X}}{n}=\frac{29}{6}=4.83$.

Adding two points to each score has also added 2 points to the mean, from M=2.83 to M=4.83.

MULTIPLYING OR DIVIDING EACH SCORE BY A CONSTANT
If every score in a distribution is multiplied by (or divided by) a constant value, the mean will change in the same way.

The following table shows how a sample of n=5 scores measured in inches would be transformed to a set of scores measures in centimeters. (Note that one inch equals 2.54 centimeters).

 Measurements transformed from inches to centimeters Original Measurement in Inches Conversion to Centimeters (multiply by 2.54) 10 25.40 9 22.86 12 30.48 8 20.32 11 27.94

The first column shows the original scores that total $\Sigma\text{X}=50$ with M=10 inches. In the second column, each of the original scores has been multiplied by 2.54 (to convert from inches to to centimiters) and the resulting values total $\Sigma\text{X}=127$, with M=25.4. Multiplying each score by 2.54 has also caused the mean to be multiplied by 2.54.