Start with the definition for the variance Equation 1, below. Expand the expression for squaring the distance of a term from the mean Equation 2, below. Now separate the individual terms of the equation the summation operator distributes over the terms in parentheses, see Equation 3, above.
Next, we can simplify the second and third terms in Equation 3. Finally, from Equation 4, you can see that the second and third terms can be combined, giving us the result we were trying to prove in Equation 5. We'll construct a table to calculate the values. You can use a similar table to find the variance and standard deviation for results from your experiments.
A histogram showing the number of plants that have a certain number of leaves. All plants have a different number of leaves ranging from 3 to 8 except for 2 plants that have 4 leaves. The difference between the highest number of leaves and lowest number of leaves is 5 so the data has relative low variance. All plants have different number of leaves ranging from 1 to The difference between the plant with the highest number of leaves and the lowest number of leaves is 10, so the data has relatively high variance.
The variance and the standard deviation give us a numerical measure of the scatter of a data set. These measures are useful for making comparisons between data sets that go beyond simple visual impressions.
The symbols also change to reflect that we are working on a sample instead of the whole population:. But hang on Imagine you want to know what the whole country thinks This is the essential idea of sampling. To find out information about the population such as mean and standard deviation , we do not need to look at all members of the population; we only need a sample.
Have a play with this at Normal Distribution Simulator. This is the formula for Standard Deviation:. Work out the Mean the simple average of the numbers 2. Then for each number: subtract the Mean and square the result 3. Then work out the mean of those squared differences. My Cart Check Out Login. Feedback Form. EdD Assistant Professor Clinical Laboratory Science Program University of Louisville Louisville, Kentucky June A simulated experiment Calculation of the mean of a sample and related statistical terminology Scores, Mean, Deviation scores First moment, Sum of squares Variance, Standard deviation Calculation of the mean of the means of samples or standard error of the mean Mean of means, Deviations or errors Sum of squares, variance of means Standard deviation of means, standard error of the mean Sample distribution of means Why are the standard error and the sampling distribution of the mean important?
Important statistical properties Important laboratory applications References Self-assessment exercises About the Author Mean or average The previous lesson described the calculation of the mean, SD, and CV and illustrated how these statistics can be used to describe the distribution of measurements expected from a laboratory method. A simulated experiment Consider the situation where there are patients available and you want to estimate the mean for that population. Calculation of the mean of a sample and related statistical terminology We will begin by calculating the mean and standard deviation for a single sample of patients.
Column A provides the individual values or scores are used to calculate the mean. Deviation scores. Column B represents the deviation scores, X-Xbar , which show how much each value differs from the mean.
In lesson four we called these the difference scores. They are also sometimes called errors as will be seen later in this lesson. First moment. The sum of the deviation scores is always zero. This zero is an important check on calculations and is called the first moment. The moments are used in the Pearson Product Moment Correlation calculation that is often used with method comparison data. Sum of squares. SS represents the sum of squared differences from the mean and is an extremely important term in statistics.
The sum of squares gives rise to variance. The first use of the term SS is to determine the variance. Variance for this sample is calculated by taking the sum of squared differences from the mean and dividing by N Standard deviation.
The variance gives rise to standard deviation. The second use of the SS is to determine the standard deviation. Laboratorians tend to calculate the SD from a memorized formula, without making much note of the terms.
Calculation of the mean of the means of samples the standard error of the mean Now let's consider the values for the twelve means in the small container. Remember that Column A represents the means of the 12 samples of which were drawn from the large container. Deviations or errors. Column C shows the squared deviations which give a SS of Variance of the means. Following the prior pattern, the variance can be calculated from the SS and then the standard deviation from the variance.
Mathematically, it is SS over N. Standard deviation of the means, or standard error of the mean. Continuing the pattern, the square root is extracted from the variance of 8.
This standard deviation describes the variation expected for mean values rather than individual values, therefore, it is usually called the standard error of the mean , the sampling error of the mean , or more simply the standard error sometimes abbreviated SE. Mathematically it is the square root of SS over N; statisticians take a short cut and call it s over the square root of N. Sampling distribution of the means.
If from the prior example of patient results, all possible samples of were drawn and all their means were calculated, we would be able to plot these values to produce a distribution that would give a normal curve.
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