Standard Deviation Normal Distribution Normal Distribution A common type of distribution where more values are closer to the mean, and as you move away from the mean, there are less data. Normal Distribution The more data collected the histogram will start to form a very distinct shape. The Bell Curve The distribution will be very bell shaped. The Bell Curve Very specific percentages of the area are located within one and two standard deviations of the mean. Percentages About 68% of the data will lie within one standard deviation from the mean. M ± SD (Likely that data will lie in this interval) About 95% of the data will lie within two standard deviations from the mean. M ± 2SD (Very likely that data will lie in this interval) Considered Not Likely that data will lie outside the 95%. Example 1 Suppose the number of songs on students iPods was recorded and the mean was 160 with a standard deviation of 35. 68% of the students should have songs between what values? 95% of the students should have songs between what values? Example 2 There are 240 smokers at PA. A sample was taken and the mean number of smokes per day was 12 with a standard deviation of 3. What percentage of students smoke between 6 and 18 smokes per day. Using the sample, how many students would you expect to smoke between 9 and 15 smokes per day? Significant Differences If a piece of data lies within 95% of the data we say that this is not a significant difference from the mean. If it lies outside the 95%, this is considered to be a significant difference. Significant Difference - Example Paula checks three samples of wheel bearings. The first is assumed to have a mean lifetime of 2000 h and a standard deviation of 400 h. The second and third boxes are unlabelled. She selects an item from each box The sample from box 2 fails after 1300 h. The sample from box 3 fails after 2900 h. Is either of the findings significant? Confidence Intervals We can accurately say that the population mean is within 95% of the sample mean. Confidence Intervals Example: Dana conducted a survey on how much money students typically spend per week on lunch. His sample of 40 randomly chosen students gave a mean of $15.25. Dana knows that the standard deviation is $2.00. What is the population mean. Confidence Intervals Example: $15.25 + 2($2.00) = $19.25 $15.25 – 2($2.00) = $11.25 Dana can be 95% confident that the population mean is between $11.25 and $19.25.