To calculate the standard deviation for this data set, we follow these steps:

Step 1: Organize the Data

From the table, we have:

• Value 4 appears 1 time
• Value 5 appears 3 times
• Value 6 appears 1 time

Step 2: Calculate the Mean

First, we find the mean of the data.

The formula for the mean ($\mu$) is:

$\mu = \frac{\sum (x \cdot f)}{N}$

Where:

• $x$ is the value
• $f$ is the frequency of each value
• $N$ is the total number of observations

Here, the sum of values times their frequencies is:

$(4 \cdot 1) + (5 \cdot 3) + (6 \cdot 1) = 4 + 15 + 6 = 25$

The total number of observations is:

$1 + 3 + 1 = 5$

So, the mean is:

$\mu = \frac{25}{5} = 5$

Step 3: Calculate the Variance

Variance ($\sigma^2$) is calculated using the formula:

$\sigma^2 = \frac{\sum f(x - \mu)^2}{N}$

Now, calculate $(x - \mu)^2$ for each value:

• For $x = 4$: $(4 - 5)^2 = 1$
• For $x = 5$: $(5 - 5)^2 = 0$
• For $x = 6$: $(6 - 5)^2 = 1$

Now multiply each squared deviation by its frequency:

• $1 \cdot 1 = 1$
• $3 \cdot 0 = 0$
• $1 \cdot 1 = 1$

Sum them up:

$1 + 0 + 1 = 2$

Now divide by the total number of observations (5):

$\sigma^2 = \frac{2}{5} = 0.4$

Step 4: Calculate the Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance:

$\sigma = \sqrt{0.4} \approx 0.632$

Step 5: Round to the Nearest Hundredth

Rounded to the nearest hundredth, the standard deviation is:

$\boxed{0.63}$

Final Answer:

The standard deviation of the data set is 0.63.

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Would you like more details or have any questions?

Related Questions:

1. How do you interpret the standard deviation in the context of this data?
2. How is variance different from standard deviation?
3. What happens to the standard deviation if we add a constant value to all the data points?
4. How do you calculate the mean and variance for grouped data?
5. Why is the square of the deviation used instead of the deviation itself?

Tip:

When dealing with data sets, the standard deviation provides insight into the spread of the data. A larger standard deviation means the values are more spread out from the mean, and a smaller one indicates they are more clustered around the mean.