To find the range and standard deviation of the data given in the frequency table, let's go through the steps carefully.

Step 1: Find the Range

The range is the difference between the highest and lowest values in the data set.

• Lowest value (min): 1
• Highest value (max): 20

So, the range is: $\text{Range} = 20 - 1 = 19$

Step 2: Find the Standard Deviation

To calculate the standard deviation, we need to find the mean (average), the squared deviations from the mean, and the weighted average of these squared deviations.

1. Calculate the Mean

The mean ($\bar{x}$) is given by: $\bar{x} = \frac{\sum (x \cdot f)}{\sum f}$ where $x$ is each data value, and $f$ is the frequency of each value.

Using the frequency data: $\sum (x \cdot f) = (1 \cdot 7) + (2 \cdot 3) + (12 \cdot 3) + (13 \cdot 5) + (16 \cdot 1) + (18 \cdot 1) + (20 \cdot 1)$ $= 7 + 6 + 36 + 65 + 16 + 18 + 20 = 168$

The total frequency, $\sum f = 21$.

So, the mean is: $\bar{x} = \frac{168}{21} = 8$

2. Calculate the Variance

The variance ($\sigma^2$) is calculated by finding the weighted average of the squared deviations from the mean: $\sigma^2 = \frac{\sum f \cdot (x - \bar{x})^2}{\sum f}$

Now, let's calculate $(x - \bar{x})^2$ for each value of $x$, multiply by $f$, and sum up.

$x$	$f$	$x - \bar{x}$	$(x - \bar{x})^2$	$f \cdot (x - \bar{x})^2$
1	7	$1 - 8 = -7$	49	$7 \cdot 49 = 343$
2	3	$2 - 8 = -6$	36	$3 \cdot 36 = 108$
12	3	$12 - 8 = 4$	16	$3 \cdot 16 = 48$
13	5	$13 - 8 = 5$	25	$5 \cdot 25 = 125$
16	1	$16 - 8 = 8$	64	$1 \cdot 64 = 64$
18	1	$18 - 8 = 10$	100	$1 \cdot 100 = 100$
20	1	$20 - 8 = 12$	144	$1 \cdot 144 = 144$

$\sum f \cdot (x - \bar{x})^2 = 343 + 108 + 48 + 125 + 64 + 100 + 144 = 932$

The variance is: $\sigma^2 = \frac{932}{21} \approx 44.38$

3. Calculate the Standard Deviation

The standard deviation ($\sigma$) is the square root of the variance: $\sigma = \sqrt{44.38} \approx 6.66$

Summary

• Range: 19
• Standard Deviation: 6.66 (rounded to two decimal places)

Would you like more details on any step or have any additional questions?

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1. What is the difference between population and sample standard deviation?
2. How do you interpret the range and standard deviation in a data set?
3. Why is standard deviation more informative than range in some cases?
4. How would changing one data point affect the standard deviation?
5. Can you calculate variance without first finding the mean?

Tip: The range gives a quick sense of the spread, while the standard deviation provides a more detailed measure of how much the data varies around the mean.