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Standard Error of a mean

Q:        A certain soft-drinks manufacturer found that the standard deviation in the weight of soda-filled cans coming off the production line is 0.006lb. It is company policy to calculate descriptive statistics for the mass (and other attributes) of samples comprising 36 filled cans. The mean weight for one such sample was found to be 0.817lb. What is the error associated with that value?
A:        If we were to draw 36-can samples from the population of all the cans filled at the factory, the sample means would be normally distributed, regardless of the population distribution. However, if we were to compare the deviation between the population mean and all the means from hundreds of 36-can samples, that standard deviation would be the same as that between the different sample means. This relationship gives rise to the formula below: where StDev is the population standard deviation, n is the sample size.
For our example, Standard deviation for frequency distributions
Q:        A sample of 50 6th grade students was asked to take an IQ test, and their scores are shown in the frequency distribution below. Calculate the standard deviation of their IQ scores.

 IQ Score 71-79 80-88 89-97 98-106 107-115 116-124 125-133 Frequency 2 8 10 12 6 8 4

A:        Since we are given frequency classes, we must first find the mid-point of each class in order to calculate the standard deviation. Additionally, the formula we will use must take into account the frequencies for each midpoint value based on the class frequencies above. That formula is: where f is the frequency and x is each midpoint value
The midpoint values are as shown in the table below. The mean is calculated from the midpoints and their respective frequencies; x 75 84 93 102 111 120 129 748.57 337.0896 87.6096 0.1296 74.6496 311.1696 709.6896 1497.14 2696.717 876.096 1.5552 447.8976 2489.357 2838.758 216.95 14.7292

As shown above, the standard deviation in IQ scores is 14.73
Remember that standard deviation is a measure of variability, however with frequency distributions since information has already been “lost” when grouping individual data points in frequency intervals, the standard deviation from such  a distribution should not be compared to that of a population where all the individual data-points are known (and used to calculate the standard deviation).

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