To calculate the standard deviation of Leah's friends' responses, we follow these steps:
- Find the mean (average) of the data.
- Calculate the variance.
- Take the square root of the variance to get the standard deviation.
Step 1: Calculate the Mean
The data points are: 1, 1, 2, 3, 3.
Mean (μ) = (Sum of all data points) / (Number of data points)
\[ \text{Mean} = \frac{1 + 1 + 2 + 3 + 3}{5} = \frac{10}{5} = 2 \]
Step 2: Calculate the Variance
Variance (σ²) is calculated using the formula:
\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \]
where \(x_i\) is each data point, μ is the mean, and N is the number of data points.
Now we calculate \((x_i - \mu)^2\) for each data point:
- For 1: \((1 - 2)^2 = (-1)^2 = 1\)
- For 1: \((1 - 2)^2 = (-1)^2 = 1\)
- For 2: \((2 - 2)^2 = (0)^2 = 0\)
- For 3: \((3 - 2)^2 = (1)^2 = 1\)
- For 3: \((3 - 2)^2 = (1)^2 = 1\)
Now, sum these squared differences:
\[ 1 + 1 + 0 + 1 + 1 = 4 \]
Now, divide by the number of data points (N = 5):
\[ \sigma^2 = \frac{4}{5} = 0.8 \]
Step 3: Calculate the Standard Deviation
Finally, take the square root of the variance:
\[ \sigma = \sqrt{0.8} \approx 0.8944 \]
However, it's important to note that if you are calculating the sample standard deviation (instead of population standard deviation), the denominator would be \(N - 1\) (which is 4):
\[ \sigma^2_{sample} = \frac{4}{4} = 1 \] \[ \sigma_{sample} = \sqrt{1} = 1 \]
So, if you were looking for the standard deviation as if it were a sample, the answer would be 1.
If considering the data as a whole population, your earlier computation (0.8944) is indeed correct. Please clarify if you're looking for sample or population standard deviation!