This test-taking strategy lets you organize your thoughts and mark relevant information in a question clearly. This particular test-taker also underlined the same information that we circled in red above and wrote out the mean, standard deviation, p, and z-score needed to complete the problem. Let’s see how the student in this example used that formula to complete the problem: Written out with the values for this problem, that becomes: We already have the standard deviation, so we can plug both values into this formula for calculating percentile: Once we use the standard deviation to find the 70 th percentile, we can use that answer to solve parts b and c.įirst, we need to get the z-score for 70 percent-another calculation which involves the standard deviation. The red circle marks the most important information you need for this problem. This question asks a student to apply the concept of standard deviation in context to determine other information about the tire treads. Take, for instance, this question from the FRQ portion of the 2009 AP® Stats exam. Often, this means using a given standard deviation to calculate another value in a different formula. The questions on the test will ask you to demonstrate your knowledge of standard deviation and interpret it in the context of a practical problem. On the AP® Statistics test, you will be given all the relevant standard deviation formulas on the AP® Stats formula sheet. Standard Deviation on the AP® Statistics Test This graph tells us that 34.1% of this data set falls between -1 and 1 standard deviation from the mean, while a mere 0.1% falls outside of -3 and 3. The points to its left are -1, -2, and -3 standard deviations from the mean, and vice versa on the right. In this figure, the x-axis represents the difference in standard deviations from the mean, while the y-axis represents the percentiles of the data set. Standard deviation can also be expressed graphically: Image Source: Wikimedia Commons The standard deviation in their response times would give valuable insight into how erratic drivers become when sleep-deprived. The control group is well rested while the 3 experimental groups have had 6, 4, and 2 hours of sleep respectively. Say, for instance, that you are testing response times of participants in a driving simulation. This is particularly useful if you are attempting to reproduce your results in a scientific study. It also allows you to compare standard deviation in results between different population groups. This allows you to compare results within a population group. Interpreting the Standard DeviationĪ high standard deviation generally means that the data points are widely scattered from the average while a low standard deviation means that the data points are closer to the mean. A lower-case n refers to the sample population while a capital N refers to the total population and n – 1 adjusts for the difference between the sample and the whole. The main difference here, apart from the use of the s for sample standard deviation and the x-bar symbol for sample mean, is the n – 1. If you need to find the standard deviation of a sample mean, refer to this formula: That’s the basic formula for standard deviation. That means that each Smith child is an average distance of 4.45 years away from the mean age of all the Smith children. The formula for finding the mean of a data set can also be expressed as: We have five values (for five Smith kids), so our N= 5.ĩ.2, then, is our mean, or µ. Then divide your result by the number of values in your data set, or N. To calculate the mean, add each number in the data set: To find the standard deviation from the mean, we first need to know what our mean, or average, is. Our data set is the five children’s ages: Let’s say we want to find the standard deviation from the mean age of the siblings. In our example, the Smith family has five children. First Step: Calculating the Mean of a Data Set Then we’ll look at an example from the AP® Statistics test. This crash course will take you through how to calculate and interpret standard deviation. The nuts and bolts of the equation are fairly simple-it just has a lot of different components to consider. Another way to think of it is to ask, “How much do the values in this data set deviate from the mean value?” Standard deviation is used to test variability in statistics by calculating the average distance from the mean of all the values in a data set.
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