7.1.6. What are outliers in the data? The data set of N = 90 ordered observations as shown below is examined for outliers: The computations are as follows: Median = (n+1)/2 largest data point = the average of the 45th and 46th ordered points = (559 + 560)/2 = 559.5 Lower quartile = .25(N+1)th ordered point = 22.75th ordered point = 411 + .75(436-411) = 429.75 Upper quartile = .75(N+1)th ordered point = 68.25th ordered point = 739 +.25(752-739) = 742.25 Interquartile range = 742.25 - 429.75 = 312.5 Lower inner fence = 429.75 - 1.5 (312.5) = -39.0 Upper inner fence = 742.25 + 1.5 (312.5) = 1211.0 Lower outer fence = 429.75 - 3.0 (312.5) = -507.75 Upper outer fence = 742.25 + 3.0 (312.5) = 1679.75 From an examination of the fence points and the data, one point (1441) exceeds the upper inner fence and stands out as a mild outlier; there are no extreme outliers.
Quartiles Quartiles are the values that divide a list of numbers into quarters. First put the list of numbers in order Then cut the list into four equal parts The Quartiles are at the "cuts" Like this: Example: 5, 8, 4, 4, 6, 3, 8 Put them in order: 3, 4, 4, 5, 6, 8, 8 Cut the list into quarters: And the result is: Quartile 1 (Q1) = 4 Quartile 2 (Q2), which is also the Median, = 5 Quartile 3 (Q3) = 8 Sometimes a "cut" is between two numbers ... the Quartile is the average of the two numbers. Example: 1, 3, 3, 4, 5, 6, 6, 7, 8, 8 The numbers are already in order In this case Quartile 2 is half way between 5 and 6: Quartile 1 (Q1) = 3 Quartile 2 (Q2) = 5.5 Quartile 3 (Q3) = 7 Interquartile Range The "Interquartile Range" is from Q1 to Q3: To calculate it just subtract Quartile 1 from Quartile 3, like this: Example: The Interquartile Range is: Box and Whisker Plot You can show all the important values in a "Box and Whisker Plot", like this: A final example covering everything: Put them in order: Cut it into quarters:
Standard Deviation The standard deviation of a probability distribution is defined as the square root of the variance where is the mean, is the second raw moment, and denotes the expectation value of . is therefore equal to the second central moment (i.e., moment about the mean), The square root of the sample variance of a set of values is the sample standard deviation The sample standard deviation distribution is a slightly complicated, though well-studied and well-understood, function. However, consistent with widespread inconsistent and ambiguous terminology, the square root of the bias-corrected variance is sometimes also known as the standard deviation, of a list of data is implemented as StandardDeviation[list]. Physical scientists often use the term root-mean-square as a synonym for standard deviation when they refer to the square root of the mean squared deviation of a quantity from a given baseline. , and To find the standard deviation range corresponding to a given confidence interval, solve (5) for , giving
How to Read and Use a Box-and-Whisker Plot The box-and-whisker plot is an exploratory graphic, created by John W. Tukey, used to show the distribution of a dataset (at a glance). Think of the type of data you might use a histogram with, and the box-and-whisker (or box plot, for short) could probably be useful. The box plot, although very useful, seems to get lost in areas outside of Statistics, but I’m not sure why. It could be that people don’t know about it or maybe are clueless on how to interpret it. Reading a Box-and-Whisker Plot Let’s say we ask 2,852 people (and they miraculously all respond) how many hamburgers they’ve consumed in the past week. Take the top 50% of the group (1,426) who ate more hamburgers; they are represented by everything above the median (the white line). Find Skews in the Data The box-and-whisker of course shows you more than just four split groups. Want to learn more about making data graphics?
Normal Distribution Data can be "distributed" (spread out) in different ways. But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this: A Normal Distribution The "Bell Curve" is a Normal Distribution. Many things closely follow a Normal Distribution: heights of people size of things produced by machines errors in measurements blood pressure marks on a test We say the data is "normally distributed". Quincunx Standard Deviations The Standard Deviation is a measure of how spread out numbers are (read that page for details on how to calculate it). When you calculate the standard deviation of your data, you will find that (generally): Example: 95% of students at school are between 1.1m and 1.7m tall. Assuming this data is normally distributed can you calculate the mean and standard deviation? The mean is halfway between 1.1m and 1.7m: Mean = (1.1m + 1.7m) / 2 = 1.4m Standard Scores How far is 1.85 from the mean?
Excel Box and Whisker Diagrams (Box Plots) Box and Whisker Charts (Box Plots) are commonly used in the display of statistical analyses. Microsoft Excel does not have a built in Box and Whisker chart type, but you can create your own custom Box and Whisker charts, using stacked bar or column charts and error bars. This tutorial shows how to make box plots, in vertical or horizontal orientations, in all modern versions of Excel. In its simplest form, the box and whisker diagram has a box showing the range from first to third quartiles, and the median divides this large box, the “interquartile range”, into two boxes, for the second and third quartiles. Sample Data and Calculations To play along at home in Excel 2007 or 2010, download the workbook Excel_2007_Box_Plot_Workbook.xlsx. Let’s use the following simple data set for our tutorial. All of these values are positive. First, insert a bunch of blank rows, and set up a range for calculations. First, compute some simple statistics, such as the count, mean, and standard deviation.
Correlation When two sets of data are strongly linked together we say they have a High Correlation. The word Correlation is made of Co- (meaning "together"), and Relation Correlation is Positive when the values increase together, and Correlation is Negative when one value decreases as the other increases Like this: Correlation can have a value: 1 is a perfect positive correlation 0 is no correlation (the values don't seem linked at all) -1 is a perfect negative correlation The value shows how good the correlation is (not how steep the line is), and if it is positive or negative. Example: Ice Cream Sales The local ice cream shop keeps track of how much ice cream they sell versus the temperature on that day, here are their figures for the last 12 days: And here is the same data as a Scatter Plot: We can easily see that warmer weather leads to more sales, the relationship is good but not perfect. In fact the correlation is 0.9575 ... see at the end how I calculated it. Correlation Is Not Good at Curves Where:
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