If the arithmetic mean of an ungrouped data is 19.2 and their variance is 15.13, find their root mean square. Round your answer to 2 decimal places.
The correct answer and explanation is:
Correct Answer: 19.58
Explanation:
The root mean square (RMS) is a measure of the magnitude of a set of numbers, combining both the average and the variation. For ungrouped data, the formula for root mean square is: RMS=μ2+σ2\text{RMS} = \sqrt{\mu^2 + \sigma^2}
Where:
- μ\mu is the arithmetic mean of the data.
- σ2\sigma^2 is the variance of the data.
In this problem:
- The arithmetic mean μ=19.2\mu = 19.2
- The variance σ2=15.13\sigma^2 = 15.13
Step 1: Square the mean: μ2=(19.2)2=368.64\mu^2 = (19.2)^2 = 368.64
Step 2: Add the variance: μ2+σ2=368.64+15.13=383.77\mu^2 + \sigma^2 = 368.64 + 15.13 = 383.77
Step 3: Take the square root: RMS=383.77≈19.58\text{RMS} = \sqrt{383.77} \approx 19.58
So, the root mean square of the data is 19.58 when rounded to two decimal places.
The root mean square value is always greater than or equal to the arithmetic mean. This is because the RMS takes into account the variability of the data. If there is no variation (that is, all data points are the same), then the RMS equals the mean. As the spread of the data increases, the RMS increases as well.
RMS is useful in many contexts such as physics (especially when dealing with alternating current), statistics, and data science because it represents the magnitude of values in a dataset regardless of sign, and provides a more complete picture than the mean alone. In this case, since the variance is relatively low compared to the square of the mean, the RMS is only slightly higher than the arithmetic mean.