Questions: Below is a sample dataset detailing different nutritional characteristics of cereal. I paired down the data set from 80 different cereals to 15 to make you calculations easier. Choose ONE characteristic (Calories, Protein, Fat, Sodium Fiber, Carbohydrates, Sugar, Potassium, Rating) and answer the following questions: 1. State which characteristic you chose. Is this data qualitative or quantitative? 2. Find the mean, median, and mode. 3. Find the range and describe the spread of the distribution. 4. Are there any obvious outliers? Why or why not?

Solution Steps

The characteristic chosen for analysis is calories. This data is quantitative as it represents numerical values.

To calculate the mean (\( \mu \)), we use the formula:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{1640}{15} = 109.33 \]

Thus, the mean of calories is:

\[ \text{Mean of calories: } 109.33 \]

Next, we find the median. The sorted data is:

\[ \text{Sorted data: } [50, 80, 100, 110, 110, 110, 110, 110, 110, 110, 120, 120, 120, 120, 160] \]

Using the formula for the rank of the median:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (15 + 1) = 8.0 \]

The quantile is at position 8, which corresponds to the value:

\[ \text{Median of calories: } 110 \]

For the mode, since the most frequently occurring value is:

\[ \text{Mode of calories: } 110 \]

The range of calories is calculated as:

\[ \text{Range} = \max(x) - \min(x) = 160 - 50 = 110 \]

To describe the spread of the distribution, we calculate the variance (\( \sigma^2 \)) and standard deviation (\( \sigma \)):

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} = 535.24 \]

\[ \sigma = \sqrt{535.24} = 23.14 \]

Thus, we have:

\[ \text{Variance of calories: } 535.24 \] \[ \text{Standard Deviation of calories: } 23.14 \]

Final Answer