the data set is 31,25,28,33,35,37,37,36,38,36,31,30,34,40,41,48,45,46,44,39.calculate the standard deviation of 35.65 ,show simple workings

To calculate the standard deviation of a data set, we can follow these steps:

### Step 1: Compute the Mean (Average)
The mean \(\mu\) is the sum of all the data values divided by the number of values. Let's calculate the mean for the data set:

\[
\text{Mean} (\mu) = \frac{31 + 25 + 28 + 33 + 35 + 37 + 37 + 36 + 38 + 36 + 31 + 30 + 34 + 40 + 41 + 48 + 45 + 46 + 44 + 39}{20}
\]

\[
\mu = \frac{744}{20} = 37.2
\]

### Step 2: Compute the Differences from the Mean
Next, we subtract the mean from each data point to find the difference:

\[
\begin{aligned}
31 - 37.2 & = -6.2 \\
25 - 37.2 & = -12.2 \\
28 - 37.2 & = -9.2 \\
33 - 37.2 & = -4.2 \\
35 - 37.2 & = -2.2 \\
37 - 37.2 & = -0.2 \\
37 - 37.2 & = -0.2 \\
36 - 37.2 & = -1.2 \\
38 - 37.2 & = 0.8 \\
36 - 37.2 & = -1.2 \\
31 - 37.2 & = -6.2 \\
30 - 37.2 & = -7.2 \\
34 - 37.2 & = -3.2 \\
40 - 37.2 & = 2.8 \\
41 - 37.2 & = 3.8 \\
48 - 37.2 & = 10.8 \\
45 - 37.2 & = 7.8 \\
46 - 37.2 & = 8.8 \\
44 - 37.2 & = 6.8 \\
39 - 37.2 & = 1.8 \\
\end{aligned}
\]

### Step 3: Square the Differences
Square each of the differences calculated in step 2:

\[
\begin{aligned}
(-6.2)^2 & = 38.44 \\
(-12.2)^2 & = 148.84 \\
(-9.2)^2 & = 84.64 \\
(-4.2)^2 & = 17.64 \\
(-2.2)^2 & = 4.84 \\
(-0.2)^2 & = 0.04 \\
(-0.2)^2 & = 0.04 \\
(-1.2)^2 & = 1.44 \\
(0.8)^2 & = 0.64 \\
(-1.2)^2 & = 1.44 \\
(-6.2)^2 & = 38.44 \\
(-7.2)^2 & = 51.84 \\
(-3.2)^2 & = 10.24 \\
(2.8)^2 & = 7.84 \\
(3.8)^2 & = 14.44 \\
(10.8)^2 & = 116.64 \\
(7.8)^2 & = 60.84 \\
(8.8)^2 & = 77.44 \\
(6.8)^2 & = 46.24 \\
(1.8)^2 & = 3.24 \\
\end{aligned}
\]

### Step 4: Compute the Mean of the Squared Differences
Now, calculate the mean of these squared differences. Sum up the squared values:

\[
38.44 + 148.84 + 84.64 + 17.64 + 4.84 + 0.04 + 0.04 + 1.44 + 0.64 + 1.44 + 38.44 + 51.84 + 10.24 + 7.84 + 14.44 + 116.64 + 60.84 + 77.44 + 46.24 + 3.24 = 724.08
\]

Now, divide by the number of data points (20):

\[
\frac{724.08}{20} = 36.204
\]

### Step 5: Take the Square Root
Finally, take the square root of the result to find the standard deviation:

\[
\sqrt{36.204} \approx 6.02
\]

Thus, the standard deviation of the data set is approximately **6.02**.

### Comparison with Given Standard Deviation (35.65)
The calculated standard deviation is much smaller than the provided value of 35.65. This suggests the provided standard deviation does not correspond to this data set.

---

Would you like to see more details or have any questions?

Here are 5 related questions:
1. What is the variance of this data set?
2. How does the standard deviation compare to the range of the data set?
3. How is standard deviation related to the mean?
4. What does a high standard deviation indicate about a data set?
5. Can you calculate the z-scores for each data point using this standard deviation?

**Tip:** The larger the standard deviation, the more spread out the data points are around the mean.

The data set is 31, 25, 28, 33, 35, 37, 37, 36, 38, 36, 31, 30, 34, 40, 41, 48, 45, 46, 44, 39. Calculate the standard deviation of 35.65, show simple workings.

Learn how to calculate the standard deviation for the data set: 31, 25, 28, 33, 35, 37, 37, 36, 38, 36, 31, 30, 34, 40, 41, 48, 45, 46, 44, 39. This problem covers key statistical concepts, including mean, variance, and standard deviation. Suitable for high school students (Grades 9-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation