Problem 29 - Algebra 2

Question

Jackson \(\qquad\) Washington \(\qquad\) King 170, 165, 140, 188, 195 \(\qquad\) 144, 177, 215, 225, 197 \(\qquad\) 166, 175, 196, 206, 219 Find the standard deviation of the weights for King High.

Step-by-Step Solution

Answer

The standard deviation of the weights for King High is approximately 19.51.

Step 1: List the Weights

Identify the weights given for King High: 166, 175, 196, 206, 219.

Step 2: Calculate the Mean

To find the mean, sum the weights and divide by the number of weights. Calculate as follows: \( \frac{166 + 175 + 196 + 206 + 219}{5} = \frac{962}{5} = 192.4 \).

Step 3: Find Each Deviation from the Mean

Subtract the mean from each weight: \(166 - 192.4 = -26.4\), \(175 - 192.4 = -17.4\), \(196 - 192.4 = 3.6\), \(206 - 192.4 = 13.6\), \(219 - 192.4 = 26.6\).

Step 4: Square Each Deviation

Square each deviation from the mean: \((-26.4)^2 = 696.96\), \((-17.4)^2 = 302.76\), \((3.6)^2 = 12.96\), \((13.6)^2 = 184.96\), \((26.6)^2 = 707.56\).

Step 5: Calculate the Mean of the Squared Deviations

Sum the squared deviations and divide by the number of weights: \( \frac{696.96 + 302.76 + 12.96 + 184.96 + 707.56}{5} = \frac{1905.2}{5} = 381.04 \). This is the variance.

Step 6: Find the Standard Deviation

Take the square root of the variance: \( \sqrt{381.04} \approx 19.51 \) (rounded to two decimal places).

Key Concepts

Understanding Variance

Variance is a crucial measure in statistics that enables us to understand how data is dispersed around the mean. It is essentially the average of the squared differences from the mean, providing insights into the variability within a data set. When the variance is large, it indicates that the data points are spread out widely from the mean. In contrast, a smaller variance suggests that the data points are closer to the mean.

To calculate variance, follow these steps:

• First, determine the mean of your data set.
• Next, find the deviation of each data point from the mean by subtracting the mean from each data point.
• Square each of these deviations to eliminate negative values.
• Finally, calculate the average of these squared deviations, which results in the variance.

Understanding variance is fundamental for gaining insights into data variability, helping to assess the predictability and reliability of data.

Calculating Mean

The mean is a simple yet powerful statistical concept that provides the average of a set of numbers. It is calculated by adding up all the numbers in a data set and then dividing the sum by the total number of values. The mean offers a central value around which other data values cluster and is often referred to as the "average."

Here's how to calculate the mean:

• Add together all the numbers in your data set.
• Count how many numbers are in the set.
• Divide the sum of the numbers by the quantity of values.

The mean can provide a quick snapshot of the data's central tendency, valuable for comparing different data sets or establishing a benchmark. However, it's important to note that the mean can be heavily affected by outliers, which are unusually high or low values in your data set.

Deviation from the Mean

Deviation from the mean indicates how much a data point differs from the average of the data set. Each deviation tells us how far, and in what direction, a data point lies in relation to the mean. Calculating deviations is essential for further statistical analysis, such as finding the variance or standard deviation.

To find deviation from the mean:

• Determine the mean of your data set.
• Subtract this mean from each data point to find the deviation for each.

This step is vital as deviations reveal the distribution of data points. While individual deviations might indicate positive or negative differences, when squared (as in calculating variance), they provide valuable insights into the overall spread of the data. Deviations highlight how data points fluctuate around the mean, signifying variability within the set.