The variance of a sample of 169 observations equals 576. The standard deviation of the sample equals
a. 13
b. 24
c. 576
d. 28,461
The correct answer and explanation is :
To find the standard deviation of a sample when the variance is given, use this basic relationship:
$$
\text{Standard Deviation} = \sqrt{\text{Variance}}
$$
In the question, the variance of a sample of 169 observations is 576. Therefore:
$$
\text{Standard Deviation} = \sqrt{576} = 24
$$
✅ Correct Answer: b. 24
Explanation (Approx. 300 Words):
Understanding the concepts of variance and standard deviation is essential in statistics, particularly when dealing with variability or spread in a data set.
- Variance measures how far each number in a data set is from the mean and therefore from every other number in the set. It gives a general idea of the spread or dispersion of the data.
- Standard deviation, on the other hand, is simply the square root of the variance. It is more intuitive than variance because it is expressed in the same unit as the original data (e.g., dollars, meters, etc.), while variance is in squared units.
In this question, you’re given the sample variance of 576. To find the sample standard deviation, you take the square root:
$$
\sqrt{576} = 24
$$
The number of observations, 169, does not directly affect this calculation because you’re already provided the computed sample variance. That number would only be relevant if you were calculating variance from raw data using the formula:
$$
s^2 = \frac{\sum (x_i – \bar{x})^2}{n – 1}
$$
Where:
- $s^2$ is the sample variance,
- $x_i$ are the sample values,
- $\bar{x}$ is the sample mean,
- $n$ is the sample size.
Since the variance has already been calculated, you just take the square root to find the standard deviation.
Common Mistakes:
- Choosing 576 as the standard deviation (that’s the variance, not its root).
- Choosing 13 or 28,461, which are distractors likely meant to mislead those who confuse square roots or miscalculate.