Covariance is a statistical quantity used to gauge a particular kind of relationship between two ordered sets of data. In mathematical terms, the covariance can be calculated as the difference between the average of the products of paired values from each set and the product of the average values of the two sets. The TI-83 Graphing Calculator can facilitate the entry of ordered lists of data and perform some statistical analyses, but lacks a single command to calculate the covariance of two lists of numbers. You must use a series of commands to find covariance using the TI-83.

Press the "STAT" button, then the "1" button to reveal the Stat List Editor. Use the arrow buttons to navigate around the lists. Find or create two empty lists to store your data into, or clear out the first two default lists (named "L1" and "L2") by pressing "STAT", then "4" (to input "ClrList"), then "2nd", then "1", then ",", then "2nd", then "2", and finally "ENTER". Press "STAT", then "1" to return to the Stat List Editor and find your empty lists.

Navigate with the arrow buttons to highlight the first entry of an empty list, then input the first number using the number keys and press "ENTER". The next entry will be automatically selected. Proceed to input all the numbers for that list, pressing "ENTER" between each number. When the first list is complete, use the right or left arrow button to navigate to the first entry of the next empty list. Fill this list as you did the first. These two lists will be treated as a set of paired numbers, and so should be the same length. Make note of the names of the lists you used, as shown at the top of their respective columns.

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Press "STAT", then the right arrow button to reveal the "STAT CALC" menu. Press "2" to input the "2-Var Stats" command. If you used the default lists "L1" and "L2", simply press enter to execute the command. Otherwise, you must input the name of one list, a comma, and the name of the other list before pressing "ENTER". To input the name of one of the default lists ("L1" through "L6"), simply press "2nd" then press one the number buttons from 1 to 6. If you used a list with a different name, press "2nd", then "STAT" to reveal the custom list names, then select the name with the arrow buttons and press "ENTER" to add that list to the command. When both list names have been added to the command, separated by a comma, press "ENTER" to execute the command. The screen will fill with the results of various statistical calculations if the command is successful. We will use a few of these calculated quantities to find the covariance between our lists.

Press "VARS", then "5", then the right arrow button, then "5" to insert the first part of the covariance formula: the sum of the pairwise products from the list entries. Next, press "รท" to indicate division, then "VARS", then "5", then "1" to insert "n", the number of elements in each list. You've now input an expression for the average of the pairwise products. Next, press "-" to input a subtraction operator, then "VARS", then "5", then "2", then "VARS", then "5", then "5". You should now see the complete covariance formula, with the product of the averages of the lists (shown as x and y with bars on top) being subtracted from the quotient of the sum of products term and the number of elements in the lists. Press "ENTER" to perform the calculation and display the covariance.

#### Tip

If you perform this calculation frequently, consider authoring a program for the TI-83 to facilitate the process. Refer to chapter 16 of the TI-83 Guidebook (see References) for information about programming. If you can transfer programs from your computer to the TI-83, you may also find free software on the Internet to help calculate covariance with the TI-83.