A researcher has collected the following sample data: 3, 5, 12, 3, 2. the mean of the sample is 5. Compute for the value of standard deviation.
The Correct Answer and Explanation is :
The standard deviation of the given sample data ([3, 5, 12, 3, 2]) is approximately 4.06.
Explanation:
Standard deviation measures how much the data values deviate from the mean. A higher standard deviation indicates greater variability in the dataset.
Step 1: Compute the Mean
The mean (average) is calculated as:
[
\text{Mean} = \frac{\sum X_i}{n}
]
Given the data: (3, 5, 12, 3, 2),
[
\text{Mean} = \frac{3 + 5 + 12 + 3 + 2}{5} = \frac{25}{5} = 5
]
Step 2: Compute Each Value’s Deviation from the Mean
We subtract the mean from each data point:
[
(3 – 5) = -2, \quad (5 – 5) = 0, \quad (12 – 5) = 7, \quad (3 – 5) = -2, \quad (2 – 5) = -3
]
Step 3: Square Each Deviation
[
(-2)^2 = 4, \quad (0)^2 = 0, \quad (7)^2 = 49, \quad (-2)^2 = 4, \quad (-3)^2 = 9
]
Step 4: Compute the Variance
Since this is a sample, we use sample variance formula, dividing by (n-1) instead of (n):
[
s^2 = \frac{\sum (X_i – \bar{X})^2}{n – 1}
]
[
s^2 = \frac{4 + 0 + 49 + 4 + 9}{5 – 1} = \frac{66}{4} = 16.5
]
Step 5: Compute the Standard Deviation
The standard deviation ((s)) is the square root of the variance:
[
s = \sqrt{16.5} \approx 4.06
]
Thus, the standard deviation of the sample is 4.06.