what is teh standard deviation of 1, 10, 50, 60, 100
The correct answer and explanation is:
Correct Answer:
The standard deviation of the numbers 1, 10, 50, 60, 100 is approximately 40.09.
Explanation:
Standard deviation is a measure of how spread out the numbers in a data set are. A low standard deviation means the values are close to the mean, while a high standard deviation means they are more spread out. The formula for standard deviation of a sample is: s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n – 1}}
Where:
- xix_i is each value
- xˉ\bar{x} is the mean of the values
- nn is the number of values
Step 1: Find the mean xˉ=1+10+50+60+1005=2215=44.2\bar{x} = \frac{1 + 10 + 50 + 60 + 100}{5} = \frac{221}{5} = 44.2
Step 2: Find the squared differences from the mean
- (1 – 44.2)² = 1876.64
- (10 – 44.2)² = 1170.24
- (50 – 44.2)² = 33.64
- (60 – 44.2)² = 249.64
- (100 – 44.2)² = 3115.24
Step 3: Find the sum of the squared differences Sum=1876.64+1170.24+33.64+249.64+3115.24=6445.4\text{Sum} = 1876.64 + 1170.24 + 33.64 + 249.64 + 3115.24 = 6445.4
Step 4: Divide by (n – 1), which is 4 6445.44=1611.35\frac{6445.4}{4} = 1611.35
Step 5: Take the square root 1611.35≈40.09\sqrt{1611.35} \approx 40.09
The final answer, 40.09, shows that the values are spread out quite far from the average. This is largely due to the wide range between the smallest (1) and largest (100) numbers. Standard deviation is useful in statistics for understanding how much variation there is in a dataset.