Question

The standard deviation of the numbers \(24,48,64,36,83\) is ........

Short answer

The standard deviation is approximately 20.76.

Step-by-step solution

Calculate the Mean

First, find the mean (average) of the numbers. Add all the numbers together and then divide by the number of elements.\[\text{Mean} = \frac{24 + 48 + 64 + 36 + 83}{5}\]\[\text{Mean} = \frac{255}{5} = 51\]

Find the Variance

Next, calculate the variance by finding the sum of the squared differences of each number from the mean, then divide by the number of observations (since we are dealing with a sample, usually we'd use \(n-1\) but assume here we're calculating population standard deviation with \(n=5\)). \[\text{Variance} = \frac{(24 - 51)^2 + (48 - 51)^2 + (64 - 51)^2 + (36 - 51)^2 + (83 - 51)^2}{5}\]Breaking it down:- \((24 - 51)^2 = 729\)- \((48 - 51)^2 = 9\)- \((64 - 51)^2 = 169\)- \((36 - 51)^2 = 225\)- \((83 - 51)^2 = 1024\)\[\text{Variance} = \frac{729 + 9 + 169 + 225 + 1024}{5} = \frac{2156}{5} = 431.2\]

Calculate the Standard Deviation

Finally, take the square root of the variance to find the standard deviation.\[\text{Standard Deviation} = \sqrt{431.2}\]\(\text{Standard Deviation} \approx 20.76\)

Key concepts

Mean Calculation

In statistical analysis, calculating the mean is often the first step. It's crucial for understanding the average value of a data set. The mean is also known as the average, and it is obtained by adding all the values together and then dividing the sum by the number of values. In this exercise, we have five numbers: 24, 48, 64, 36, and 83. By adding these numbers, we get a total of 255. To find the mean, we divide this total by 5, which is the number of data points. Thus, the mean calculation is as follows: - Sum of numbers = 24 + 48 + 64 + 36 + 83 = 255 - Number of data points = 5 - Mean = 255/5 = 51 Understanding the mean helps in getting a sense of the overall size of the numbers in the dataset.

Variance Calculation

Variance is an essential concept in statistics as it shows how much the numbers in the set differ from the mean. Higher variance indicates that the numbers are more spread out from the mean. To calculate variance, subtract the mean from each data point and square the result to remove negative signs. Then, sum these squared differences. Finally, divide this summed value by the number of observations. In our case:- Subtracting the mean (51) from each number: - - 24: (24-51) = -27, and - 48: (48-51) = -3, etc.- Squaring these: - - (-27)^2 = 729, and - (-3)^2 = 9, etc.Now, accrue these squared values: 729, 9, 169, 225, and 1024, resulting in a sum of 2156. Divide this sum by 5 to find the variance,\[\text{Variance} = \frac{2156}{5} = 431.2\] This number shows the average of the squared differences from the mean.

Statistical Analysis

Statistical analysis involves summarizing data to uncover patterns or insights. The process generally includes calculating various metrics such as mean, variance, and standard deviation. In our exercise, we interpret these statistics to understand our dataset more clearly. 1. **Mean** shows the average value among the data points.2. **Variance** tells us how spread out the numbers are from the mean.3. **Standard Deviation** offers a measure of dispersion by taking the square root of the variance.In this exercise, after calculating variance, we find the standard deviation by taking the square root of 431.2. Thus,\[\text{Standard Deviation} = \sqrt{431.2} \approx 20.76\]This comprehensively indicates how much the numbers differ from the mean. Together, these statistics offer a clear image of data behavior and form the basis for more advanced statistical interpretations.