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Quick Summary: Highlights of Sections 9.1 and 9.2 Measures of Central Tendency The MODE is defined to be the most frequent number in a set of numbers. The MEDIAN is defined to be the middle number (in value) in a set of numbers. [If there is no 'single' middle number (in value), the median is defined as the average of those 2 middle numbers.] The MEAN is defined to be the ordinary arithmetic average of a set of numbers. [Sum the numbers and divide by the number of numbers in the set.] Measures of Variation The SAMPLE RANGE is defined to be the Largest Value - Smallest Value in the sample. The SAMPLE VARIANCE (s2) is defined as: The SAMPLE STANDARD DEVIATION (s) is defined to be the square root of the Sample Variance: Shortcut formula for calculating the SAMPLE VARIANCE/STANDARD DEVIATION (by hand) The formula above lets you to calculate the mean or standard deviation by hand. In practice, you will want to use your own calculator (one allowed in the MTLC for your test). Below is how this calculation is done on the CASIO fx-260 calculator. (Other calculators will be at least somewhat different.) Detailed instructions below show how to use the CASIO fx-260 calculator to do these calculations. Working with GROUPED DATA Range of scores (x interval) 12 - 17 18 - 23 24 - 29 30 - 35 Example of a FREQUENCY DISTRIBUTION Frequency (f) 3 6 4 2 Example of a FREQUENCY HISTOGRAM Calculating a GROUPED MEAN BY HAND USING THE CASIO fx—260 CALCULATOR 1. Find the midpoint of each interval (xmp) 1. Find the midpoint of each interval (xmp) 2. Multiply each midpoint by its frequency (f) 2. Enter each midpoint into the Stats Mode 3. Add up all the resulting products of the calculator & then press 4. Divide this sum by n (which is just the sum of the f ’s) SEE THE ‘STATS ON THE CASIO fx-260’ SHEET FOR MORE DETAILED INSTRUCTIONS. Calculating a GROUPED VARIANCE/STANDARD DEVIATION BY HAND USING THE CASIO fx—260 CALCULATOR 1. Calculate the square of each midpoint 1. Find the midpoint of each interval (xmp) 2. Multiply each square by its frequency 2. Enter each midpoint into the Stats Mode 3. Add up these products of the calculator & then press 4. Plug into the formula for s2 5. Take the square root to get s (if you want s instead of s2) SEE THE ‘STATS ON THE CASIO fx-260’ SHEET FOR MORE DETAILED INSTRUCTIONS. The procedure above shows you how to calculate a group mean or standard deviation by hand. In practice, you will want to use your own calculator (one allowed in the MTLC for your test). Below is how this calculation is done on the CASIO fx-260 calculator. (Other calculators will be at least somewhat different.) Instructions below show how to use the CASIO fx-260 calculator to do these calculations. Chebyshev’s Theorem: For any set of numbers, a fraction of at least of them will be within K standard deviations of the mean. So, for example, for k=2 standard deviations of the mean, at least of the numbers in the data set will be within 2 standard deviations of the mean. STATS on the CASIO fx-260 SOLAR Calculator Wakes up the calculator…but it also clears out any old data previously stored. Puts calculator into Stats (SD) mode [indicated by ‘SD’ in upper right of display]. Clears old stats data (but unlike the ON button, it leaves you in Stat mode) Key in a number & press M+ to add it to the dataset. Once your dataset has been entered, this calculates the mean. Once your dataset has been entered, this calculates the (sample) standard deviation. Note: Once a number is entered, pressing enters it again. Shortcut to enter the same number multiple times: 7x8 automatically enters the number 7 eight times (the long way: 7 7 7 7 7 7 7 7 ) Note: If you enter a number in error, you can take it out by re-entering the number and pressing SHIFT . Note: You can use SHIFT to display how many numbers have been entered so far.