Problem 11 - Algebra 2

Question

Find the variance and standard deviation of each set of data to the nearest tenth. {13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 67, 56, 34, 99, 44, 55}

Step-by-Step Solution

Answer

Variance: 633.0; Standard Deviation: 25.2.

Step 1: Find the Mean

To calculate the mean, sum all the data points and divide by the total number of data points. The sum of the data set is:\[13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 23 + 67 + 56 + 34 + 99 + 44 + 55 = 531\]There are 16 numbers in this data set, so the mean is:\[\frac{531}{16} = 33.1875\]

Step 2: Find the Squared Differences

Subtract the mean from each data point, and square the result to find the squared differences. For example, for the first data point:\[(13 - 33.1875)^2 = 408.140625\]Repeat this calculation for every data point.

Step 3: Sum the Squared Differences

Now, sum all the squared differences:\[408.140625 + 367.765625 + 326.390625 + 285.015625 + 243.640625 + 202.265625 + 160.890625 + 119.515625 + 78.140625 + 54.765625 + 1149.390625 + 515.015625 + 0.640625 + 4327.015625 + 117.390625 + 480.515625 = 10127.75\]

Step 4: Calculate the Variance

To find the variance, divide the sum of the squared differences by the total number of data points:\[\text{Variance} = \frac{10127.75}{16} = 633.0\]

Step 5: Calculate the Standard Deviation

The standard deviation is the square root of the variance. Thus, it is calculated as:\[\sqrt{633.0} \approx 25.2\]

Step 6: Wrap-up

The variance is 633.0 and the standard deviation is approximately 25.2 when rounded to the nearest tenth.

Key Concepts

Standard Deviation

The standard deviation is a measure that tells us how spread out the numbers in a data set are. It’s a way to find out if your data points are close to the mean (average) or if they are spread out over a wider range. When the standard deviation is small, data points are close to the mean.
Conversely, a large standard deviation indicates that the data points are spread out. To calculate it, we first find the variance. Then, we take the square root of the variance value to get the standard deviation. In our example, the variance calculated was 633.0, so the standard deviation came out to be approximately 25.2. This means there is a moderate spread of numbers from the average value in our data set.

Mean Calculation

The mean is often referred to as the "average." It is calculated by adding up all the numbers in a data set and then dividing by the total count of numbers. It's a useful measure of central tendency, as it gives us a central value that represents the overall data set.
For example, if we want to find the mean of a set of numbers like {13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 67, 56, 34, 99, 44, 55}, we first sum these numbers, giving us 531. Then we divide by the total number of elements in the set, which is 16. Hence, the mean calculation is \( \frac{531}{16} \), which equals 33.1875. This tells us that, on average, each number is about 33.2 when rounded to the nearest tenth.

Squared Differences

After we calculate the mean, the next step in variance and standard deviation calculation involves finding the squared differences. Each data point is compared to the mean by subtracting the mean from it. This gives us the deviation for each number.
Instead of working with negative and positive numbers, we then square each of these deviations (because a squared number is always positive). Squaring each difference gives us results that reflect the amount of variation from the mean without dealing with the direction of deviation. For instance, for the first data point 13, the squared difference is \( (13 - 33.1875)^2 = 408.140625 \). We follow the same approach for all other data points.

Statistical Analysis

Statistical analysis involves using mathematical computations to interpret and analyze data sets. Tools like mean, variance, and standard deviation help us quantify patterns and reveal insights within the data.
By calculating the standard deviation and variance, we learn about the dispersion and consistency of the data. These analyses are crucial for making informed decisions based on empirical data. For instance, a high standard deviation amongst exam scores might prompt a teacher to adjust their teaching strategy. Thus, statistical analysis serves as a powerful means to understand and make sense of various forms of data, facilitating better decision-making across numerous fields.