Population and Sample Standard Deviations. You examine the numerical relationship between the population standard deviation and the sample standard deviation computed from the same data. This relationship is helpful when the computer or statistical calculator being used has a built-in program for sample standard deviation but not for population standard deviation. Consider a data set with m observations. If the data are sample data, you compute the sample standard deviation, s, whereas if the data are population data, you compute the population standard deviation, σ. a. Derive a mathematical formula that gives σ in terms of s when both are computed for the same data set. (Hint: First note that, numerically, the values of x and μ are identical. Consider the ratio of the defining formula for σ to the defining formula for s.) b. Verify that your formula in part (a) works for each of the three data sets c. Suppose that a data set consists of 15 observations. You compute the sample standard deviation of the data and obtain s = 38.6. Then you realize that the data are actually population data and that you should have obtained the population standard deviation instead. Use your formula from part (a) to obtain σ.
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
Population and Sample Standard Deviations. You examine the numerical relationship between the population standard deviation and the sample standard deviation computed from the same data. This relationship is helpful when the computer or statistical calculator being used has a built-in program for sample standard deviation but not for population standard deviation.
Consider a data set with m observations. If the data are sample data, you compute the sample standard deviation, s, whereas if the data are population data, you compute the population standard deviation, σ.
a. Derive a mathematical formula that gives σ in terms of s when both are computed for the same data set. (Hint: First note that, numerically, the values of x and μ are identical. Consider the ratio of the defining formula for σ to the defining formula for s.)
b. Verify that your formula in part (a) works for each of the three data sets
c. Suppose that a data set consists of 15 observations. You compute the sample standard deviation of the data and obtain s = 38.6. Then you realize that the data are actually population data and that you should have obtained the population standard deviation instead. Use your formula from part (a) to obtain σ.
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