Listed below are measured amounts of caffeine​ (mg per 12oz of​ drink) obtained in one can from each of 14 brands. Find the​ range, variance, and standard deviation for the given sample data. Include appropriate units in the results. Are the statistics representative of the population of all cans of the same 14 brands​ consumed?

4848

3434

3939

3333

00

40

53

58

47

55

41

55

0

0

Given the measured amounts of caffeine (in mg per 12 oz of drink) for 14 different brands:

\[ 48, 34, 39, 33, 0, 40, 53, 58, 47, 55, 41, 55, 0, 0 \]

### Step 1: Find the Range
The range is the difference between the maximum and minimum values in the dataset.

- Minimum value \( \text{min} = 0 \) mg
- Maximum value \( \text{max} = 58 \) mg

\[
\text{Range} = \text{max} - \text{min} = 58 - 0 = 58 \text{ mg}
\]

### Step 2: Find the Variance
To find the variance, we use the formula for sample variance \( s^2 \):

\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2
\]

Where:
- \( n = 14 \) (number of data points),
- \( x_i \) represents each data point, and
- \( \bar{x} \) is the sample mean.

#### Calculate the mean \( \bar{x} \):

\[
\bar{x} = \frac{\sum x_i}{n} = \frac{48 + 34 + 39 + 33 + 0 + 40 + 53 + 58 + 47 + 55 + 41 + 55 + 0 + 0}{14} = \frac{503}{14} \approx 35.93 \text{ mg}
\]

#### Calculate each squared deviation \( (x_i - \bar{x})^2 \):

\[
\sum (x_i - \bar{x})^2 = (48 - 35.93)^2 + (34 - 35.93)^2 + \dots + (0 - 35.93)^2 \approx 148.37 + 3.72 + \dots + 1291.26 = 6560.5
\]

#### Calculate the sample variance \( s^2 \):

\[
s^2 = \frac{6560.5}{14-1} = \frac{6560.5}{13} \approx 504.65 \text{ mg}^2
\]

### Step 3: Find the Standard Deviation
The standard deviation is the square root of the variance.

\[
s = \sqrt{s^2} \approx \sqrt{504.65} \approx 22.46 \text{ mg}
\]

### Interpretation of Results
- **Range**: 58 mg
- **Variance**: \( \approx 504.65 \text{ mg}^2 \)
- **Standard Deviation**: \( \approx 22.46 \text{ mg} \)

### Are the Statistics Representative of the Population?
The statistics calculated are based on the given sample data of caffeine content in one can from each of 14 brands. However, to determine if these statistics are representative of the entire population (all cans from these 14 brands), we would need to consider the following:
- **Sampling Method**: If these 14 measurements are randomly selected and representative of all cans, then the statistics can be considered fairly representative. However, if the measurements are not representative (e.g., they are from a specific batch or time period), then the statistics might not reflect the overall population.
-

Learn how to calculate the range, variance, and standard deviation of caffeine content in 14 different brands. Includes step-by-step calculations and interpretations based on sample data. Suitable for high school students studying descriptive statistics.

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