What is the relationship between the population standard deviation and the sample standard deviation?

My question is similar pnd1987's question. I wish to use a standard deviation in order to appraise the repeatability of a measurement. Suppose I'm measuring one stable thing over and over. A perfect measuring instrument (with a perfect operator) would give the same number over and over. Instead there is variation, and let's assume there's a normal distribution about the mean.

We'd like to appraise the measurement repeatability by the SD of that normal distribution. But we take just N measurements at a time, and hope the SD of those N can estimate the SD of the normal distribution. As N increases, sampleSD and populationSD both converge to the distribution's SD, but for small N, like 5, we get only weak estimates of the distribution's SD. PopulationSD gives an obviously worse estimate than sampleSD, because when N=1 populationSD gives the ridiculous value 0, while sampleSD is correctly indeterminate. However, sampleSD does not correctly estimate the disribution's SD. That is, if we measure N times and take the sampleSD, then measure another N times and take the sampleSD, over and over, and average all the sampleSDs, that average does not converge to the distribution's SD. For N=5, it converges to around 0.94× the distribution SD. (There must be a little theorem here.) SampleSD doesn't quite do what it is said to do.

If the measurement variation is normally distributed, then it would be very nice to know the distribution's SD. For example, we can then determine how many measurements to take in order tolerate the variation. Averages of N measurements are also normally distributed, but with a standard deviation 1/sqrt(N) times the original distribution's.

Note added: the theorem is not so little -- Cochran's Theorem

As a business owner, you are constantly figuring out what your current customers want and what your potential customer needs. The data can be tracked in a variety of ways, from polls and surveys to interviews and historical research. However, the tool used to put this data into results, standard deviation, can be used in several ways, depending on the type of results you're seeking.

Tip

Standard deviation is the measurement of spread in a data set. It can be used to help decide the best choice from among several options. The difference between sample and population standard deviation is the data set.

What Is Standard Deviation?

Standard deviation is the dispersion between two or more data sets. For example, if you were designing a new business logo and you presented four options to 110 customers, the standard deviation would indicate the number who chose Logo 1, Logo 2, Logo 3 and Logo 4. The standard deviation is calculated by finding the mean, calculating the variance and taking the square root of the variance.

Find the Mean, Variance and Standard Deviation

The mean is the average of the numbers in the dataset. Keeping with the logo example, let's say 25 people liked Logo 1, 30 people liked Logo 2, 35 people like Logo 3 and 20 people like Logo 4. The mean would be the result of (25 + 30 + 35 + 20) / 4 or 27.5 rounded to 28. To find the variance, first find the difference between the mean and each set of data. So for the logos, the differences would be -3 (25-28 ), 2 (30 - 28), 7 (35 - 28) and -8 (20 - 28) respectively.

The next step is to square the differences, which equals 9, 4 and 49 and 64. Now you have to find the average of the squared numbers to get the variance which is 31.5 rounded up to 32 ((9 + 4 + 49 + 64) / 4). Finally, calculate the standard deviation by finding the square root of the variance, which is 5.6 or 6.

How Is This Useful?

Knowing the standard deviation can help you determine which option is best for your business. Thinking back to the logo, the mean was 28. A standard deviation of 6 means the logos whose votes were within 6 points of the mean is the most popular choice. So, as for the logos, more people like Logos 1 and 2 than they liked 3 and 4.

Sample Standard Deviation

The above calculated is the population standard deviation. It dealt with a specific set of data. However, if you wanted to determine the standard deviation of a large population you would use the sample standard deviation. The only difference in the calculation is that you subtract 1 from the number used to calculate the variance.

So, going back to the logos, instead of dividing the squares of the differences by four, you would divide them by three (9 + 4 + 49 + 64) / 3 = 42. then find the square root, which is 6.

When to Use Sample or Population?

If you want to measure your current customer's reactions or opinions, stick to population standard deviation, since that is a more quantifiable number. However, if you are experimenting with new ways to attract new customers, then a sample deviation would be better, because you can include more variables such as gender, age and geographic locations.

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