When we talk about statistics, especially how data spreads out, there are three important terms to know: range, variance, and standard deviation. Let’s break them down to make them easier to understand.

Range

The range is the easiest of the three terms. It tells us how far apart the highest and lowest numbers in a set are.

To find the range, you subtract the smallest number from the largest number.

For example, if we have these numbers: 3, 7, 5, and 10, the range is:

$$\text{Range} = \text{Maximum} - \text{Minimum} = 10 - 3 = 7$$

So, the range is 7. This means that the numbers spread out 7 units from the lowest to the highest.

Variance

Next, we have variance. This term helps us understand how different the numbers are from the average (or mean).

To calculate variance, follow these steps:

1. First, find the mean:

   $$\text{Mean} = \frac{4 + 6 + 8}{3} = 6$$

2. Then, find the squared differences from the mean:

   • For 4: $(4 - 6)^2 = 4$
   • For 6: $(6 - 6)^2 = 0$
   • For 8: $(8 - 6)^2 = 4$

3. Finally, average those squared differences:

   $$\text{Variance} = \frac{4 + 0 + 4}{3} = \frac{8}{3} \approx 2.67$$

Standard Deviation

Now, let’s talk about standard deviation. This is just the square root of the variance. It helps us understand how spread out the data is in a simpler way, using the same units as the original numbers.

For our variance of about 2.67, the standard deviation would be:

$$\text{Standard Deviation} = \sqrt{2.67} \approx 1.63$$

Summary

So, to sum it up:

• Range tells us how widely the data is spread.
• Variance gives us the average amount the numbers differ from the mean.
• Standard Deviation helps us see how much the numbers usually vary, using the same units as the original numbers.

These ideas are really useful when we want to understand how data is spread out in statistics!