Question

9 multiple choice 1 point
seven doctors offices were surveyed to see how many patients they saw on an average day. the results of the survey are shown below.
51, 35, 98, 26, 47, 88, 66
find the standard deviation of the set of data.
58.714
28.443
713.905
26.719
809.004

Explanation:

Step1: Calculate the mean ($\mu$)

The formula for the mean is $\mu = \frac{\sum_{i = 1}^{n} x_i}{n}$, where $x_i$ are the data points and $n$ is the number of data points.
Here, $n = 7$, and the data points are 51, 35, 98, 26, 47, 88, 66.
$\sum_{i = 1}^{7} x_i = 51 + 35 + 98 + 26 + 47 + 88 + 66 = 411$
$\mu = \frac{411}{7} \approx 58.714$

Step2: Calculate the squared differences from the mean

For each data point $x_i$, calculate $(x_i - \mu)^2$:

• $(51 - 58.714)^2 \approx (-7.714)^2 \approx 59.506$
• $(35 - 58.714)^2 \approx (-23.714)^2 \approx 562.354$
• $(98 - 58.714)^2 \approx (39.286)^2 \approx 1543.409$
• $(26 - 58.714)^2 \approx (-32.714)^2 \approx 1069.206$
• $(47 - 58.714)^2 \approx (-11.714)^2 \approx 137.218$
• $(88 - 58.714)^2 \approx (29.286)^2 \approx 857.669$
• $(66 - 58.714)^2 \approx (7.286)^2 \approx 53.086$

Step3: Calculate the variance ($\sigma^2$)

The formula for the variance (population variance, since we are dealing with all 7 data points as the population here) is $\sigma^2 = \frac{\sum_{i = 1}^{n} (x_i - \mu)^2}{n}$
$\sum_{i = 1}^{7} (x_i - \mu)^2 \approx 59.506 + 562.354 + 1543.409 + 1069.206 + 137.218 + 857.669 + 53.086 = 4282.448$
$\sigma^2 = \frac{4282.448}{7} \approx 611.778$

Step4: Calculate the standard deviation ($\sigma$)

The standard deviation is the square root of the variance: $\sigma = \sqrt{\sigma^2}$
$\sigma = \sqrt{611.778} \approx 24.734$ (Wait, this doesn't match the options. Wait, maybe we should use sample standard deviation? Wait, the problem says "the set of data" - if it's a sample, we divide by $n - 1$. Let's check that.

Wait, maybe I made a mistake. Let's recalculate with sample standard deviation (divide by $n - 1 = 6$).

$\sum_{i = 1}^{7} (x_i - \mu)^2 \approx 4282.448$ (same as before)
Sample variance $s^2 = \frac{4282.448}{6} \approx 713.741$
Sample standard deviation $s = \sqrt{713.741} \approx 26.719$ (Ah, this matches one of the options. So probably the problem considers it a sample. Let's recheck the steps.)

Wait, let's recalculate the mean correctly: 51 + 35 = 86; 86 + 98 = 184; 184 + 26 = 210; 210 + 47 = 257; 257 + 88 = 345; 345 + 66 = 411. 411 / 7 = 58.714 (correct).

Squared differences:
51 - 58.714 = -7.714; squared: 59.506 (correct)
35 - 58.714 = -23.714; squared: 562.354 (correct)
98 - 58.714 = 39.286; squared: 1543.409 (correct)
26 - 58.714 = -32.714; squared: 1069.206 (correct)
47 - 58.714 = -11.714; squared: 137.218 (correct)
88 - 58.714 = 29.286; squared: 857.669 (correct)
66 - 58.714 = 7.286; squared: 53.086 (correct)
Sum: 59.506 + 562.354 = 621.86; +1543.409 = 2165.269; +1069.206 = 3234.475; +137.218 = 3371.693; +857.669 = 4229.362; +53.086 = 4282.448 (correct)

Sample variance: 4282.448 / 6 ≈ 713.741 (wait, 4282.448 / 6 = 713.741333...). Then sample standard deviation is sqrt(713.741333) ≈ 26.719 (since 26.719^2 ≈ 713.9, which is close, considering rounding errors). So the correct answer is 26.719.

Answer:

26.719 (corresponding to the option with 26.719)