Problem 53 - Thinking Mathematically

Question

Use the data \(1,2,3,5,6,7\). Without actually computing the standard deviation, which of the following best approximates the standard deviation? a. 2 b. 6 c. 10 d. 20

Step-by-Step Solution

Answer

The best approximation for the standard deviation is 2 (option a).

Step 1: Understand standard deviation

In this step, standard deviation should be understood as the average distance of each data point from the mean. From the given data set \(1,2,3,5,6,7\), we can see the numbers are relatively close to each other.

Step 2: Determine the range of the data

The range of a set is the difference between the highest and lowest values. Here this is \(7 - 1 = 6\). This already suggests that the standard deviation will not be a large number, as there is not a large spread in the data.

Step 3: Evaluate the options

Now look at the options given. Think about these in relation to the range. Here, options c and d (10 and 20) are already bigger than the total range of our data and can hence be dismissed. Between 2 and 6, since the data is quite evenly spread and not too far from its mean, 2 (option a) is a better approximation.

Key Concepts

Standard Deviation

Standard deviation is a crucial statistical tool that measures the spread or dispersion of a set of data. In simple terms, it tells you how much the values in the data set diverge from the average value—the mean. Think of it like measuring how much the members of a sports team differ in height. Small standard deviation means the heights are similar; large standard deviation means there's a big difference in heights.

When trying to approximate the standard deviation, you don't need to calculate it exactly. Instead, you can get a sense of how the values in a data set are spread around the mean. With our data set of \(1,2,3,5,6,7\), the values aren't very far apart. So, you can expect a lower standard deviation, which in this case is closer to 2 rather than 6, 10, or 20.

Statistical Range

The statistical range of a data set is found by subtracting the smallest value from the largest value. It's a quick estimate of how spread out the numbers are. Yet, keep in mind that it doesn't tell you anything about the spread of the rest of the data, only the extremes.

For example, in our data set \(1,2,3,5,6,7\) the range is \(7 - 1 = 6\). This gives us a quick snapshot of the outermost values' spread. When you're approximating the standard deviation and you find the range is small (like 6), it's a sign that the data points are generally close together, hence implying a smaller standard deviation.

Mean

In statistics, the mean is the average of all values in a data set. To calculate it, you add up all the numbers and then divide by how many there are. It's used as a measure of central tendency, representing the middle of a data set.

Using our data set \(1,2,3,5,6,7\), the mean would be calculated as \((1+2+3+5+6+7) / 6\), which simplifies to \(4\). The mean is often near the center of the range and is a reference point for measuring the standard deviation, which contextualizes how far away the rest of the numbers in the set are from this central value.

Data Spread

Data spread, or dispersion, indicates how stretched or squeezed the data is. It's all about the distance between the data points. For instance, consider a scatter of seeds—dispersion is like looking at how far apart they've landed. In stats, this concept helps to understand the variability within the data set.

In our exercise, we used the range and the mean to infer the data spread without having to delve into complex calculations. A small range, like we have with our data set, means there's not a lot of data spread, hinting that the standard deviation should be relatively small as well. This leads us to choose the correct approximation for the standard deviation, which in this case is option a: 2.