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Statistics for Business - Mean and Standard Deviation

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  • What is the formula for the mean of a data set?
    The mean is calculated as \(\bar{x} = \frac{\sum x}{n}\), where \(\sum\) x is the sum of all data values and n is the number of values.
  • How do you calculate the mean of the data set {5, 10, 14, 12, 3}?
    Sum the values: 5 + 10 + 14 + 12 + 3 = 44, then divide by 5. Mean = \(\frac{44}{5} = 8.8\).
  • What is the effect of adding an outlier to a data set on the mean?
    Adding an outlier (extreme value) can significantly change the mean, making it less representative of the data.
  • What is the formula for sample standard deviation?
    Sample standard deviation is \(s = \sqrt{\frac{1}{n-1} \sum (x - \bar{x})^2}\), measuring how spread out data values are around the mean.
  • What does a larger standard deviation indicate about a data set?
    A larger standard deviation means the data values are more spread out from the mean.
  • Calculate the deviations from the mean for the data set {5, 10, 12, 14, 3} if the mean is 8.8.
    Deviations: 5 - 8.8 = -3.8, 10 - 8.8 = 1.2, 12 - 8.8 = 3.2, 14 - 8.8 = 5.2, 3 - 8.8 = -5.8.
  • How do you calculate the squared deviations for the data set {5, 10, 12, 14, 3} with mean 8.8?
    Square each deviation: (-3.8)^2=14.44, 1.2^2=1.44, 3.2^2=10.24, 5.2^2=27.04, (-5.8)^2=33.64.
  • What is the sum of squared deviations for the data set {5, 10, 12, 14, 3} with mean 8.8?
    Sum = 14.44 + 1.44 + 10.24 + 27.04 + 33.64 = 86.8.
  • How do you calculate the sample variance from squared deviations?
    Sample variance = sum of squared deviations divided by (n - 1), where n is the number of data points.
  • Calculate the sample variance for the data set {5, 10, 12, 14, 3} with sum of squared deviations 86.8.
    Variance = \(\frac{86.8}{5-1} = 21.7\).
  • How do you find the sample standard deviation from the variance?
    Standard deviation is the square root of the variance.
  • Calculate the sample standard deviation for variance 21.7.
    Standard deviation = \(\sqrt{21.7} \approx 4.66\).
  • What is the difference between population and sample standard deviation formulas?
    Population uses \(N\) in denominator; sample uses \(n-1\) to correct bias.
  • What does the symbol \(\bar{x}\) represent in statistics?
    \(\bar{x}\) represents the sample mean, the average of sample data values.
  • What does the symbol s represent in statistics?
    s represents the sample standard deviation, a measure of data spread in a sample.
  • Why is the sample standard deviation always greater than or equal to zero?
    Because it is a square root of squared deviations, which are always non-negative.
  • What is the meaning of the summation symbol \(\sum\) in the mean formula?
    \(\sum\) means to add all the values of the variable x in the data set.
  • How do you interpret a standard deviation of zero?
    All data values are identical; there is no spread.
  • What is the first step in calculating the mean of a data set?
    Sum all the data values.
  • What is the second step in calculating the mean of a data set?
    Divide the sum by the number of data values.