Question

For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand. $$ \begin{array}{ll} {\text { Set 1 }} & {\text { Set 2 }} \\ \hline \text { a) } 4,7,7,7,10 & 4,6,7,8,10 \\ \text { b) } 100,140,150,160,200 & 10,50,60,70,110 \\ \text { c) } 10,16,18,20,22,28 & 48,56,58,60,62,70 \end{array} $$

Short answer

Set 2 has larger standard deviations for parts a and c; Set 1 has larger standard deviation for part b.

Step-by-step solution

Analyze Set a)

For Set a, observe that Set 1 and Set 2 have different spreads. Set 1 has a middle clustering (three 7s) whereas Set 2 is more evenly spread around 7. The larger spread in Set 2 suggests a larger standard deviation. Let's confirm by calculating the actual standard deviations of both sets.

Calculate Mean for Set a)

Calculate the mean of Set 1: \((4+7+7+7+10)/5 = 7\). Calculate the mean of Set 2: \((4+6+7+8+10)/5 = 7\).

Calculate Variance for Set a)

For Set 1: Variance is \(((4-7)^2 + (7-7)^2 + (7-7)^2 + (7-7)^2 + (10-7)^2)/5 = 6/5 = 1.2\).For Set 2: Variance is \(((4-7)^2 + (6-7)^2 + (7-7)^2 + (8-7)^2 + (10-7)^2)/5 = 20/5 = 4\).

Calculate Standard Deviation for Set a)

Standard deviation is the square root of variance.Set 1: \(\sqrt{1.2} \approx 1.1\).Set 2: \(\sqrt{4} = 2\). Thus, Set 2 has a larger standard deviation.

Analyze Set b)

In Set b, observe that both sets have exactly the same set of numbers but different scales. Set 2 is simply Set 1 divided by 10. Since standard deviation is also scaled by the same factor, Set 1 clearly has the larger standard deviation.

Confirm with Calculations for Set b)

Calculate the mean for both sets and find the variance similarly. You will find they both follow the same relative pattern (mean, variance division or multiplication by 10), confirming the larger standard deviation for Set 1.

Analyze Set c)

In Set c, observe that both sets also have similar internal variations, but at different scales - Set 2 numbers are multiples of Set 1's numbers. Since the higher magnitude numbers lead to a greater spread, Set 2 is expected to have the larger standard deviation.

Confirm with Calculations for Set c)

Calculate the mean and variance for both sets. Since every variance multiplication by a constant is identical for both, Set 2 will confirmatively have the larger standard deviation given its scale.

Key concepts

Variance

Variance is a fundamental statistical concept, which measures the degree of spread in a set of data. It tells us how much individual data points differ from the mean of the dataset.

- **Calculation**: To find the variance: - Calculate the mean of the data. - Subtract the mean from each data point to get the deviation from the mean. - Square each deviation to eliminate negative values. - Find the average of these squared deviations. This value is the variance!
- **Example from Exercise**: Reflecting on Set a: - For Set 1, after calculating the squared differences from the mean, the variance was found to be 1.2. - For Set 2, using the same process, the variance was determined to be 4.
- **Importance**: A higher variance indicates a larger spread of data points from the mean, and thus, suggests greater variation within the data set.

Mean

The mean is commonly known as the average. It is a measure that represents the central point of a dataset.

- **Calculation**: - Add up all the numbers in the dataset. - Divide by the total number of values to get the mean.
- **Example**: In Set a from our exercise, both Set 1 and Set 2 have the same mean of 7. This helps to easily compare other measures like variance and standard deviation between the sets.

- **Significance**: The mean provides a quick snapshot of the dataset's central tendency but doesn't provide any information about the variability of the data.

Data Distribution

Data distribution describes how data points are spread over the range of values. Understanding this helps us quantify patterns and outliers in the data.

- **Analysis**: - In the exercise, Set a displayed two different types of distributions: - Set 1 had a clustering of three identical numbers resulting in a narrower data distribution. - Set 2, with evenly spread values, demonstrated a wider distribution.
- **Patterns**: Data distribution patterns such as uniformity or clustering immediately provide insights into the characteristics of the dataset, which affects calculations of standard deviation and variance. This offers an intuitive sense of which set is more variable before performing detailed calculations.
- **Implications**: A wider distribution typically means a larger standard deviation, denoting more variability.

Statistical Comparison

Statistical comparison involves assessing multiple datasets to see how they relate in terms of variability and central tendency. Tools like mean, variance, and standard deviation are often used here.

- **Process**: - Compare the means of the datasets to determine differences in central location. However, even identical means like those in Set a do not imply identical distributions. - Use variance and standard deviation to compare variability. Larger values in these metrics often indicate more spread or inconsistency.
- **Example**: In the exercise, each part demonstrates statistical comparison: - Regardless of similar means, distinct variances reveal different spreads and therefore, potential implications on standard deviation.
- **Purpose**: The primary aim of statistical comparison is to understand underlying differences between datasets and use that understanding to apply insights in decision-making or predictions.