Problem 32 - Algebra 2

Question

For Exercises \(31-32,\) use the following bowling scores for six members of a bowling team: \(175,210,180,195,208,196 .\) What is the standard deviation of the scores?

Step-by-Step Solution

Answer

The standard deviation of the scores is 13

Step 1: Calculate the Mean

Add up all the scores and divide by the number of scores to find the mean. That is, \( \text{Mean} = \frac{175 + 210 + 180 + 195 + 208 + 196}{6} = 194 \)

Step 2: Find the Deviations

Subtract the mean from each score to find the deviation of each score. Their deviations are: \(175 - 194 = -19\), \(210 - 194 = 16\), \(180 - 194 = -14\), \(195 - 194 = 1\), \(208 - 194 = 14\), \(196 - 194 = 2\)

Step 3: Square the Deviations

Square each deviation you found. The squared deviations are: \((-19)^2 = 361\), \(16^2 = 256\), \((-14)^2 = 196\), \(1^2 = 1\), \(14^2 = 196\), \(2^2 = 4\)

Step 4: Average the Squared Deviations

Add the squared deviations together and divide by the number of scores to find the average of the squared deviations. That is, \( \text{Mean Squared Deviation} = \frac{361 + 256 + 196 + 1 + 196 + 4}{6} = 169 \)

Step 5: Calculate the standard deviation

Take the square root of the result from Step 4. That is, \( \text{Standard Deviation} = \sqrt{169} = 13 \)

Key Concepts

Understanding the Mean

The mean is a way to describe the average score in a set of numbers. It acts like the center balance point. For our bowling scores, calculating the mean involves adding all the individual scores together and then dividing by the total number of scores. This number gives us an idea of what a typical score might look like among the team members. In this case, the mean of the bowling scores is calculated by adding all the scores: 175, 210, 180, 195, 208, and 196. The sum is divided by 6, which provides the mean score of 194.
This number helps compare each member’s score to the group average, guiding us towards understanding how spread out the scores are from the center.

Calculating Deviations

Deviations are important for measuring how far each score is from the mean. Essentially, a deviation tells us how different a specific score is compared to the average. To find the deviation for each bowling score, we subtract the mean from the individual scores.

• For a score of 175, the deviation is 175 - 194 = -19.
• Similarly, for the score of 210, the deviation is 210 - 194 = 16.

By calculating these deviations, we can see whether scores are generally clustered around the mean or if they are more spread out. Negative deviations indicate scores below the mean, while positive deviations show scores above the mean.

Finding Squared Deviations

Squared deviations are used to eliminate any negative sign from deviations and emphasize larger differences from the mean. When we square each deviation, we are magnifying the larger deviations even more, which highlights outlying scores. For the bowling scores, squaring involves taking each calculated deviation and multiplying it by itself:

• For a deviation of -19, the squared deviation is (-19)^2 = 361.
• Whereas a deviation of 16 becomes 16^2 = 256.

Squaring these numbers makes all deviations positive and provides a clearer picture of variation among the scores.

Understanding Mean Squared Deviation

The mean squared deviation (also known as variance) is the average of all the squared deviations. It summarizes the extent of dispersion or spread in the data set. To find it, you add up all the squared deviations and then divide by the number of data points. In our case, the sum of squared deviations is divided by 6, yielding the mean squared deviation of 169.
This value is crucial because it simplifies understanding how spread out the scores are, giving us insight into the variability in the team's bowling performance. It acts as a baseline for calculating the standard deviation when you take its square root.