Solved STANDARD DEVIATION WORKSHEET A teachers records show ... - Free Printable

Problem Overview:

We are tasked with calculating the standard deviation for a dataset of absences for 6 students. The dataset is given as: \( \{1, 3, 4, 6, 9, 19\} \). To solve this, we will follow these steps:

1. Calculate the mean (\( \bar{x} \)) of the dataset.
2. Find the deviation from the mean for each data point.
3. Square the deviations.
4. Sum the squared deviations.
5. Calculate the variance by dividing the sum of squared deviations by \( n-1 \) (since it's a sample).
6. Find the standard deviation by taking the square root of the variance.

Let's proceed step by step.

---

Step 1: Calculate the Mean (\( \bar{x} \))

The mean is calculated as:
\[
\bar{x} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\]
The dataset is \( \{1, 3, 4, 6, 9, 19\} \).

First, find the sum:
\[
1 + 3 + 4 + 6 + 9 + 19 = 42
\]

Next, divide by the number of data points (\( n = 6 \)):
\[
\bar{x} = \frac{42}{6} = 7
\]

So, the mean is:
\[
\bar{x} = 7
\]

---

Step 2: Find the Deviation from the Mean

The deviation from the mean for each data point \( x_i \) is calculated as:
\[
\text{Deviation} = x_i - \bar{x}
\]

For each data point:
- For \( x = 1 \):
\[
1 - 7 = -6
\]
- For \( x = 3 \):
\[
3 - 7 = -4
\]
- For \( x = 4 \):
\[
4 - 7 = -3
\]
- For \( x = 6 \):
\[
6 - 7 = -1
\]
- For \( x = 9 \):
\[
9 - 7 = 2
\]
- For \( x = 19 \):
\[
19 - 7 = 12
\]

So, the deviations are:
\[
\{-6, -4, -3, -1, 2, 12\}
\]

---

Step 3: Square the Deviations

The squared deviations are calculated as:
\[
(x_i - \bar{x})^2
\]

For each deviation:
- For \( -6 \):
\[
(-6)^2 = 36
\]
- For \( -4 \):
\[
(-4)^2 = 16
\]
- For \( -3 \):
\[
(-3)^2 = 9
\]
- For \( -1 \):
\[
(-1)^2 = 1
\]
- For \( 2 \):
\[
2^2 = 4
\]
- For \( 12 \):
\[
12^2 = 144
\]

So, the squared deviations are:
\[
\{36, 16, 9, 1, 4, 144\}
\]

---

Step 4: Sum the Squared Deviations

The sum of the squared deviations is:
\[
\sum d^2 = 36 + 16 + 9 + 1 + 4 + 144
\]

Calculate the sum:
\[
36 + 16 = 52
\]
\[
52 + 9 = 61
\]
\[
61 + 1 = 62
\]
\[
62 + 4 = 66
\]
\[
66 + 144 = 210
\]

So, the sum of the squared deviations is:
\[
\sum d^2 = 210
\]

---

Step 5: Calculate the Variance

The variance (\( s^2 \)) for a sample is given by:
\[
s^2 = \frac{\sum d^2}{n-1}
\]
where \( n \) is the number of data points.

Here, \( n = 6 \), so:
\[
s^2 = \frac{210}{6-1} = \frac{210}{5} = 42
\]

---

Step 6: Find the Standard Deviation

The standard deviation (\( s \)) is the square root of the variance:
\[
s = \sqrt{s^2} = \sqrt{42}
\]

Using a calculator:
\[
\sqrt{42} \approx 6.48
\]

---

Final Answer:

The value for the standard deviation is:
\[
\boxed{6.48}
\]

---

Completed Table:

\[
\begin{array}{|c|c|c|c|}
\hline
x & \bar{x} & \text{Deviation From the Mean} & \text{Squared Deviations} \\
\hline
1 & 7 & -6 & 36 \\
3 & 7 & -4 & 16 \\
4 & 7 & -3 & 9 \\
6 & 7 & -1 & 1 \\
9 & 7 & 2 & 4 \\
19 & 7 & 12 & 144 \\
\hline
\end{array}
\]

\[
\sum d^2 = 210
\]