To calculate the standard deviation of the dataset with the given values and observations, we will follow these steps:
- Calculate the mean (average).
- Calculate the variance.
- Take the square root of the variance to find the standard deviation.
The given values and their corresponding observations are:
- Value 40 → 1 observation
- Value 50 → 2 observations
- Value 60 → 1 observation
Step 1: Calculate the Mean
First, we find the total number of observations (N): \[ N = 1 + 2 + 1 = 4 \]
Next, calculate the sum of the weighted values: \[ \text{Sum} = (40 \times 1) + (50 \times 2) + (60 \times 1) = 40 + 100 + 60 = 200 \]
Now, we can find the mean \(\mu\): \[ \mu = \frac{\text{Sum}}{N} = \frac{200}{4} = 50 \]
Step 2: Calculate the Variance
Now we will calculate the variance \(\sigma^2\): \[ \sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N} \] where \(x_i\) are the values, \(\mu\) is the mean, and \(f_i\) is the frequency (number of observations).
Calculating each term:
- For value 40: \[ (40 - 50)^2 \times 1 = (-10)^2 \times 1 = 100 \]
- For value 50: \[ (50 - 50)^2 \times 2 = (0)^2 \times 2 = 0 \]
- For value 60: \[ (60 - 50)^2 \times 1 = (10)^2 \times 1 = 100 \]
Now sum these up: \[ \sum (x_i - \mu)^2 \cdot f_i = 100 + 0 + 100 = 200 \]
Now we can calculate the variance: \[ \sigma^2 = \frac{200}{4} = 50 \]
Step 3: Calculate the Standard Deviation
Finally, we take the square root of the variance: \[ \sigma = \sqrt{50} \approx 7.07 \]
Rounding to the nearest hundredth, the standard deviation is approximately: \[ 7.07 \]
Answer
The correct answer is 7.07.