Explanation:
Step 1: Find the mean x of the data values.
X = (ix₁) / n
x = (12 + 3 + 9 + 2 + 0 + 0 + 3) / 7
x = 29 / 7
x ≈ 4.14 (rounded to two decimal places)
Step 2: Calculate the squared differences from the mean for each data value [(x₁ - x)²].
For each data value, subtract the mean and square the result.
Squared Differences:
(12 - 4.14)² ≈ 53.43
(3 - 4.14)² ≈ 1.31
(9 - 4.14)² ≈ 23.09
(2 - 4.14)² ≈ 4.34
(0 - 4.14)² ≈ 17.11
(0 - 4.14)² ≈ 17.11
(3 - 4.14)² ≈ 1.31
Step 3: Calculate the variance (s²) of the data values.
s² = ((x₁ - x)²) / (n - 1)
s² = (53.43 + 1.31 + 23.09 + 4.34 + 17.11 + 17.11 + 1.31) / (7 - 1)
s² ≈ 117.70 / 6
s² ≈ 19.62 (rounded to two decimal places)
Step 4: Calculate the standard deviation (s) of the data values.
s = √s²
s = √19.62
s ≈ 4.43 (rounded to two decimal places)
Step 5: Find the range of the data values.
Range = Maximum value - Minimum value
Range = 12 - 0
Range = 12
Therefore, the results are:
The standard deviation is approximately 4.43 (rounded to two decimal places).
The variance is approximately 19.62 (rounded to two decimal places).
The range is 12.