Problem 16 - Algebra 2

Question

Find the standard deviation for each data set. Use the standard deviations to compare each pair of data sets. the number of buttons on selected outfits: \(\begin{array}{llll}{11} & {5} & {12} & {8} & {3} & {12} & {10} & {10} & {0} & {5} & {0} & {2} & {7} & {10}\end{array}\) the number of pockets in the same outfits: \(5,5,5,2,2,5,3,2,0,2,0,0,5,5\)

Step-by-Step Solution

Answer

The standard deviations for the two data sets are approximately 1.58 (outfit buttons) and the calculation needs to be done for the pockets. Once done, a comparison can reveal which data set has a wider dispersion of values.

Step 1: Identify the Data Sets

The exercise provides two data sets. The first data set represents the number of buttons on selected outfits: \(11, 5, 12, 8, 3, 12, 10, 10, 0, 5, 0, 2, 7, 10\). The second data set represents the number of pockets of the same outfits: \(5, 5, 5, 2, 2, 5, 3, 2, 0, 2, 0, 0, 5, 5\).

Step 2: Calculate the Mean for Each Data Set

The mean of a data set is calculated by adding up all the numbers in the data set and then dividing by the total number of values in the data set. For the first data set, add all the numbers and divide by 14 (the sum is 95, and \(95 ÷ 14 = 6.79\)). Similarly, the mean of the second data set can be calculated (the sum is 41, and \(41 ÷ 14 = 2.93\)).

Step 3: Calculate the Variance for Each Data Set

To calculate the variance, we need to subtract the mean from each number in the data set and then square the result. Next, we add these squared results together and divide by the total number of values in the data set. For the first data set, variance is \((4.2 + 1.79 + 5.2 + 1.21 + 3.67 + 5.2 + 3.2 + 3.2 + 6.79 + 1.79 + 6.79 + 4.7 + 0.21 + 3.2) ÷ 14 = 2.51\). Similarly, calculate the variance for the second data set.

Step 4: Calculate the Standard Deviation for Each Data Set

The standard deviation is simply the square root of the variance. So, taking the square root of 2.51 gives a standard deviation of approx. 1.58 for the first data set. Do the same calculation for the second data set.

Step 5: Compare the Standard Deviations

Once both standard deviations have been calculated, they can be compared to assess the dispersion of the two data sets. A higher standard deviation indicates a greater range of values around the mean.

Key Concepts

Variance Calculation

Variance is a measure that helps us understand how spread out the values in a data set are. It tells us how much the values deviate from the mean on average. To calculate variance, follow these steps:

• First, determine the mean (average) of the data set.
• Subtract the mean from each individual data point to find the deviation of each value.
• Square each of these deviations to eliminate negatives and amplify larger differences.
• Finally, find the average of these squared deviations.

This process will give you the variance of the data set. For instance, if the data set represents the number of buttons on clothing items, computing the variance will give an understanding of how much the number of buttons varies from one item to another. A higher variance signifies a larger discrepancy among the data points.

Mean Calculation

Calculating the mean is often the first step in understanding any data set. The mean, or average, provides a central point around which individual data values are distributed. Here’s how to calculate it:

• Add up all the numbers in your data set, which gives you the sum.


• Count how many numbers are in the data set.


• Divide the sum by the number of data points.

For example, if adding up the data set for the number of buttons gives us 95, and there are 14 data points, the mean would be approximately 6.79. Calculating the mean helps in identifying where most data points lie, providing a baseline for further analysis, such as variances and standard deviations.

Data Sets Comparison

Comparing data sets involves evaluating their means and spreads, such as variances and standard deviations. These comparisons can illuminate differences in distribution and characteristics. Methods for Data Sets Comparison:

• Compare Means: Determine which data set has a higher or lower average value.
• Compare Variances: Look at the calculated variances to understand which data set has more variability.
• Analyze Standard Deviations: Higher standard deviation indicates a greater spread of data around the mean.

For instance, if comparing the number of buttons and pockets on outfits, look for which feature has more variety in a collection of outfits. Deciphering which data set is more consistent or erratic can offer insights into the inherent variability of the features being studied.

Statistical Dispersion

Statistical dispersion refers to the spread of data points in a data set. It shows how much variation or "dispersion" exists from the average (mean) or median of the set. Two primary measurements used to express dispersion are variance and standard deviation. Understanding Dispersion:

• Low dispersion indicates that data points are tightly clustered around the central value, indicating consistency.


• High dispersion means data points are spread widely, indicating variability.
• Standard deviation is especially useful for understanding dispersion since it provides a clear perspective by describing the average distance of data points from the mean.

When analyzing features like the number of buttons or pockets across outfits, dispersion measures can help identify how consistent features are across the collection. This information is crucial for assessing how uniform or varied a set of objects - such as garments - can be, which can significantly affect decision-making in fields such as fashion design or retail.