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### Calculating Measures of Central Tendency and Variability from Math Exam Scores

• 1.
• 2.  In order to calculate with any data, the data must be organized.  Here are the scores received by a 12th grade math class for their final exam organized from lowest score to highest score.  57 58 62 67 73 75 75 75 77 78 78 82 82 84 84 85 85 86 86 88 90 91 91 94 95
• 3.  Data set: 57 58 62 67 73 75 75 75 77 78 78 82 82 84 84 85 85 86 86 88 90 91 91 94 95  The Mode is simply the number that occurs most frequently. So in this data set it would be 75. It is possible to have two modes in which case it would be called bi-modal.  The Median is literally the middle number of the data set. If there is an odd number of observations it is the number in the middle. If it is an even number you take the two middle numbers, add them and then divide them by 2. The Median in our data set is 82 because there are 25 observations in our set and 82 is right in the middle.
• 4.  The Mean is simply an average, you only need to add up all the observations and divide it by the number of observations.  So, 57+58+62+67+73+75+75+75+77+78+78+82+82+ 84+84+85+85+86+86+88+90+91+91+94+95=1931  Then, you divide the sum by the number of observations which is 25.  1931/25=77.24 (just round to the nearest whole number for a percent)  From that it is easy to see that the average grade of the class is 77%.
• 5.  The Range is easy to calculate. Just take the highest number in the data set and subtract it from the lowest number. ◦ Data set: 57 58 62 67 73 75 75 75 77 78 78 82 82 84 84 85 85 86 86 88 90 91 91 94 95  So, for this data set the range is: 95 – 57 = 38.  Here we can say that the class grades only varied within 38 points.
• 6.  “Variance measures the deviation of the data set from its mean.”  The Variance formula is:  Don’t worry it is not as complicated as it looks.  First you take all the unique observations individually and subtract it from the mean and square that number. Afterwards you add all the results together and divide them by n-1 where n = the number of observations.  It’s easiest to do this on a table to be able to keep track of everything.
• 7. x (x – X) (x – X)² 57 57 - 77.24 = -20.24 (-20.24)² = 409.66 58 58 - 77.24 = -19.24 (-19.24)² = 370.18 62 62 - 77.24 = -15.24 (-15.24)² = 232.26 67 67 - 77.24 = -10.24 (-10.24)² = 104.86 73 73 - 77.24 = -4.24 (-4.24)² = 17.98 75 75 - 77.24 = -2.24 (-2.24)² = 5.02 77 77 - 77.24 = -0.24 (-0.24)² = 0.06 78 78 - 77.24 = 0.76 (0.76)² = 0.58 82 82 - 77.24 = 4.76 (4.76)² = 22.66 84 84 - 77.24 = 6.76 (6.76)² = 45.70 85 85 - 77.24 = 7.76 (7.76)² = 60.22 86 86 - 77.24 = 8.76 (8.76)² = 76.74 88 88 - 77.24 = 10.76 (10.76)² = 115.78 90 90 - 77.24 = 12.76 (12.76)² = 162.82 91 91 - 77.24 = 13.76 (13.76)² = 186.34 94 94 - 77.24 = 16.76 (16.76)² = 280.90 95 95 - 77.24 = 17.76 (17.76)² = 315.42
• 8.  So, the Sigma (Σ) part of the equation is just telling us to add up all the results from the final column of our table which. ◦ 409.66+370.18+232.26+104.86+17.98+5.02+5.02+5.02 +0.06+0.58+0.58+22.66+22.66+45.70+45.70+60.22+ 60.22+76.74+76.74+115.78+162.82+186.34+186.34+ 280.90+315.42 = 2868.62 ◦ Note that we had 25 observations and must add 25 numbers, the repeated numbers were not added to the table but you must add them here for the sum.
• 9.  So, now that we’ve done all that work to get that number we can finally calculate the Variance.  The n in the formula is simply the number of observations which in this case is 25.  s² = 2868.62÷(25-1) = 119.53
• 10.  In order for the Variation to be useful we must calculate the Standard Deviation.  Standard Deviation formula:  Now that we’ve done all the work this is easy. ◦ s = 119.53 = 10.93 Note-This is just the square root of the variance
• 11.  To find the Quartiles of our data we first have to go to the median of the set which is 82.  57 58 62 67 73 75 75 75 77 78 78 82 82 84 84 85 85 86 86 88 90 91 91 94 95 ◦ Now you have a lower half {57,82} and an upper half {84,95}  2nd Quartile will be at 82 ◦ The Median will always be the 2nd Quartile.
• 12.  To find the 1st and 3rd Quartiles you have to find the median of the lower and upper halves.  Since there is an even number of observations you have to take the two middle numbers add them and then divide by 2 to find the median.  Lower Half= 57 58 62 67 73 75 75 75 77 78 78 82 75+75 = 150 ÷2 = 75  Upper Half= 84 84 85 85 86 86 88 90 91 91 94 95 86+86 = 172 ÷2 = 86  1st Quartile will be between {57,82}  2nd Quartile will be at 82  3rd Quartile will be between {84,95}
• 13.  Finding the quartiles is useful for percentages. ◦ The median of the 1st Quartile = 25% ◦ The median of the 2nd Quartile = 50% ◦ The median of the 3rd Quartile = 75%  Interquartile Range shows how much the data varies from the 2nd quartile which is at 50%.  To calculate Interquartile Rage simply subtract the third quartile from the first. ◦ 86-75= 11 ◦ The Interquartile Range for our data set is 11.
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