Understanding data is important in Year 1 Math at Gymnasium, especially when we talk about things like range, variance, and standard deviation. These tools help us see how data is spread out and how it varies. This is really helpful for things like sports and fitness tracking.

Range

The range is the easiest way to see how spread out data is. You find it by subtracting the smallest value from the biggest value. Here’s how it looks:

$$\text{Range} = \text{Max} - \text{Min}$$

For example, let’s say a group of students timed their 100-meter sprints, and the times were 12.1, 11.8, 12.5, and 11.9 seconds.

To find the range, we do this:

$$\text{Range} = 12.5 - 11.8 = 0.7 \text{ seconds}$$

This means the times are pretty close together, showing that the students have similar abilities.

Variance

Next, we have variance. This tells us how far each number is from the average (mean). To find variance, we look at how much each number differs from the average, square those differences, and then find the average of those squared numbers.

The formula for variance looks like this:

$$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$$

In this formula, $x_i$ represents each score, $\mu$ is the average, and $N$ is the total number of scores.

Using our sprint times, if the average time is 12.0 seconds, we calculate the variance like this:

$$\sigma^2 = \frac{(12.1 - 12.0)^2 + (11.8 - 12.0)^2 + (12.5 - 12.0)^2 + (11.9 - 12.0)^2}{4} = 0.015$$

Standard Deviation

Now, let’s talk about standard deviation. This tells us how far each data point is from the average, on average. It’s found by taking the square root of the variance.

The formula is:

$$\sigma = \sqrt{\sigma^2}$$

So, if our variance is 0.015, we can find the standard deviation like this:

$$\sigma = \sqrt{0.015} \approx 0.123$$

Conclusion

To wrap it up, range, variance, and standard deviation are useful tools for Year 1 students at Gymnasium. They help you understand and analyze numbers better. With these tools, students can see how consistent their performance is, find any unusual results, and make smart choices about fitness and sports training. Knowing these ideas is really important for building a strong base in statistics and math in the Swedish curriculum.