In the next section of the course material and in the following activity, you will need to be familiar with a method for measuring how far a given data value is from the mean. You’ve seen before how to calculate deviation from the mean for a given data value. This method standardizes the distance in units of standard deviation above or below the mean. The calculations will be performed by hand, and they can be a little tricky. In this corequisite support activity, you’ll get some practice with them and learn where and how they can go wrong.

Measuring Distance from Mean

We’ll use a hypothetical typical arm span for Americans in our exploration of this method for measuring distance from the mean. Arm span is the distance from the tip of the middle finger on one hand to the tip of the middle finger on the other hand when one’s arms are stretched out and opened wide.

Suppose that, based on many measurements, a statistician believes that the distribution of arm spans of Americans has a mean of [latex]173.40[/latex] centimeters (cm) and a standard deviation of [latex]12.21[/latex] cm. Let’s understand that these values to pertain to the population of all American arm spans (not just a sample).

For Questions 1, 3, and 5 below, you’ll calculate arm span values that are one, two, and one-and-a-half standard deviations above and below the mean of [latex]173.40[/latex] cm. Then, you’ll carefully follow the directions in Questions 2, 4, and 6, to make the specified calculations for each of your answers to the previous question. Note that in each of these, you’ll have found the difference between the given value and the mean, and divided that distance by the standard deviation.  We call this standardizing the value, but  we’ll cover that more deeply in the next section. For now, let’s just get some practice making the necessary calculations by hand.

Standard Deviation

Now you try it. In Question 1, add or subtract the standard deviation to or from the mean to calculate the new value.

In Question 2, apply the formula [latex]\dfrac{\text{value}-\text{mean}}{\text{standard deviation}}[/latex] to each of the values you obtained in Question 1.

In Question 1 you calculated a value exactly one standard deviation above and one below the mean. In Question 2, you divided the difference between the value you calculated and the mean by the standard deviation. You should have obtained positive one and negative one for the values above and below, respectively. What does positive imply? How about negative?

Let’s try another couple of sets of questions like that to observe what’s happening.

Hopefully you obtained [latex]2[/latex] and [latex]-2[/latex] for your answers to Question 4. Have you caught on to what’s happening in these question pairs yet?  Let’s try one more pair. In Question 5, you’ll identify values one and a half standard deviations above and below the mean. Can you predict what the answers to Question 6 should be?

In Questions 2, 4, and 6, you calculated values that were a given number of standard deviations above and below the mean. You discovered when you divided the difference between the value and the mean by the standard deviation, that the result was a positive number of standard deviations (for values above the mean) or a negative number of standard deviations (for values below the mean). That is, a resulting negative can be thought of as indicating a value that lies to the left of the mean, and the positive indicates a value that lies to the right of the mean.

Standardizing a Value

A natural question to consider might be, given any value any distance from the mean in any direction, if we find the difference between the value and mean, then divide by the standard deviation, would we be able to discover the number of standard deviations any value is from the mean and whether it lies to the right or to the left? Answer Questions 7 and 8 to explore this idea.

Now let’s spend some time understanding how and in what way these kinds of calculations can go wrong.

Identifying Calculation Mistakes

Suppose the statistician making these calculations thought she was using her calculator correctly, but in three different attempts, she arrived at three different answers. The three potential answers to her computational problem are shown below in Questions 9, 10, and 11 rounded to the nearest hundredth. For each, decide if it was computed correctly or, if not, explain what went wrong. Please refer to the Order of Operations Student Resource as needed.

Now it’s your turn to try finding the mistakes. Some of the questions below may have a mistake, and some may not.

Questions 12, 13, and 14 below show three potential answers to a similar computational problem, rounded to the nearest hundredth. This time, the units of measure are included. For each, decide if it was computed correctly or, if not, explain what went wrong.

Now that you’ve had some practice making these calculations and learning how they can go wrong, it’s time to move on to the next section.