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The range is the difference between the greatest and least value. The standard deviation is the average amount by which each individual value deviates or differs from the mean.
Juniors Range: 11
Seniors Range: 17
Juniors Standard Deviation: Approximately 3.43
Seniors Standard Deviation: Approximately 5.74
Comparison: See solution.
We want to find the range, and standard deviation of the two given data sets, as well as compare our results.
Now that we found the ranges, we want to find the standard deviation. To do it, we will need the mean, so we will calculate it first.
The mean of a data set is the sum of the values divided by the total number of values in the set. Let's start by calculating the sum of the given values. Juniors:&& 19+15+20+10+14 + && 21+18+15= 132 Seniors:&& 22+19+29+32+15 + && 26+30+19= 192 There are 8 values in each set, so we have to divide their sums by 8. Juniors Mean:&& 132/8&=16.5 Seniors Mean:&& 192/8&=24
The standard deviation of a set of data is the average amount by which each individual value deviates or differs from the mean. Standard Deviation sqrt((x_1-x )^2+(x_2-x )^2+... +(x_n-x )^2/n) In the above formula, x_1, ... ,x_n are the values of the set of data, x is the mean, and n is the number of values. For this exercise, we already found the means for both sets. Juniors $x$:&& 132/8&=16.5 Seniors $x$:&& 192/8&=24 Let's use these values and apply the formula to each value in the sets. We will do it separately for each data set. Let's consider the juniors group first.
| x_n | x_n-x | (x_n-x)^2 |
|---|---|---|
| 19 | 19-16.5=2.5 | 2.5^2=6.25 |
| 15 | 15-16.5=- 1.5 | (- 1.5)^2=2.25 |
| 20 | 20-16.5=3.5 | 3.5^2=12.25 |
| 10 | 10-16.5=- 6.5 | (- 6.5)^2=42.25 |
| 14 | 14-16.5=- 2.5 | (- 2.5)^2=6.25 |
| 21 | 21-16.5=4.5 | 4.5^2=20.25 |
| 18 | 18-16.5=1.5 | 1.5^2=2.25 |
| 15 | 15-16.5=- 1.5 | (- 1.5)^2=2.25 |
| Sum of Values | = 94 | |
Finally, we need to divide by 8 and then calculate the square root. Juniors Standard Deviation: sqrt(94/8)≈ 3.43 Now, let's consider the seniors group. Remember that we need to consider the other value for x!
| x_n | x_n-x | (x_n-x)^2 |
|---|---|---|
| 22 | 22-24=- 2 | (- 2)^2=4 |
| 19 | 19-24=- 5 | (- 5)^2=25 |
| 29 | 29-24=5 | 5^2=25 |
| 32 | 32-24=8 | 8^2=64 |
| 15 | 15-24=- 9 | (- 9)^2=81 |
| 26 | 26-24=2 | 2^2=4 |
| 30 | 30-24=6 | 6^2=36 |
| 19 | 19-24=- 5 | (- 5)^2=25 |
| Sum of Values | = 264 | |
Finally, we need to divide by 8 and then calculate the square root. Seniors Standard Deviation: sqrt(264/8)≈ 5.74
The seniors range is greater than the juniors range. This means the points for seniors are more spread out. Similarly, the seniors standard deviation is greater than that of the juniors group, which means that juniors are more consistent than the seniors.