This document provides information about various statistical measures calculated from a data set of math exam scores of a 12th grade class. It defines mode as the most frequent score (75), median as the middle score (82), mean as the average score (77), range as the difference between highest and lowest scores (95-57=38), variance as a measure of deviation from the mean (119.53), standard deviation as the square root of variance (10.93), and quartiles as divisions of the data into four equal parts (1st quartile: 75, 2nd quartile: 82, 3rd quartile: 86). The interquartile range is defined as the difference between the third and first quartiles (

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.

14.  http://www.allianthawk.org/showdef.php?wo
rd=187
 http://media.pearsoncmg.com/pcp/pls_coco
_2/statistics_ap/lesson_03/index.html