It provides an estimate of the population mean for the sample using the specific measurement method. The sample mean, denoted , is calculated using the formula:. Suppose we use atomic absorbance spectroscopy to measure the total sodium content a can of soup; we perform the measurement on five separate portions of the soup, obtaining the results What is the mean value for the sodium content of the can of soup?
We also need to determine the spread of results about the mean value, in order to provide more specific information on how many significant figures we can attribute to our sample mean. We can do this by calculating the sample variance, which is the average of the squared difference between each measurement and the sample mean i. A simple justification for this is that it is impossible to estimate the measurement dispersion with a single reading — we would have to assume that the spread of results is infinitely wide.
As noted in the introduction , it is more convenient to use the standard deviation, which is simply the square root of the variance,. The advantage of using standard deviation over variance for describing your results is that s has the same units as the mean value. Use the worksheet from exercise 1 to also calculate the variance and standard deviation of the sodium values by setting up a formula.
You will need to create a column to calculate individual values of before calculating s 2 and s. Note that a single value, or a mean value without any indication of the sample variance or standard deviation, is scientifically meaningless.
Note also that the first non-zero digit of the standard deviation identifies the least significant digit of the mean. Here are the steps:. Notice that the standard deviation is a little bit larger than the average deviation which was 2. We can get a good approximation of the standard deviation by estimating the average distance from the mean.
The shaded box on the following dotplot indicates 1 SD to the right and left of the mean. The formula for the standard deviation of a data set can be described by the following expression. However, we will always use technology to perform the actual computation of the standard deviation. Before we learn to use technology to compute the standard deviation, we practice estimating it.
We can estimate standard deviation in the same ways we estimated ADM. The standard deviation is therefore:. Mean and Standard Deviation. How to Subscribe. Need multiple seats for your university or lab? Get a quote.
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