Multiple Choice
Which of the following is key to generating a bell curve in statistics?
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3. Describing Data Numerically
Standard Deviation
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Given the data set , what is the standard deviation of the data? Round your answer to the nearest whole number.
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Verified step by step guidance1
First, calculate the mean (average) of the data set. The mean \( \bar{x} \) is found by summing all the data points and dividing by the number of points: \( \bar{x} = \frac{65 + 75 + 100 + 130}{4} \).
Next, find the deviation of each data point from the mean by subtracting the mean from each value: \( (x_i - \bar{x}) \) for each \( x_i \) in the data set.
Then, square each of these deviations to get \( (x_i - \bar{x})^2 \). This step ensures all values are positive and emphasizes larger deviations.
Calculate the variance by finding the average of these squared deviations. Since this is a sample, divide the sum of squared deviations by \( n - 1 \) (where \( n \) is the number of data points): \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \).
Finally, take the square root of the variance to get the standard deviation: \( s = \sqrt{s^2} \). This gives a measure of spread in the original units of the data.
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