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Standard Error of a mean
Q: A certain softdrinks manufacturer found that the standard deviation in the weight of sodafilled 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 36can 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 36can 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 
7179 
8088 
8997 
98106 
107115 
116124 
125133 
Frequency 
2 
8 
10 
12 
6 
8 
4 
A: Since we are given frequency classes, we must first find the midpoint 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 
748.5696 
337.0896 
87.6096 
0.1296 
74.6496 
311.1696 
709.6896 

1497.139 
2696.717 
876.096 
1.5552 
447.8976 
2489.357 
2838.758 


216.9504 


14.72924 
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 datapoints are known (and used to calculate the standard deviation).