0.1  Alternative definition of the mean

It is useful to present an alternative formulation of the definition of the mean (which is also valid for the variance and standard deviation; i.e., s2 and s, not s2 and s). The usual definition can be rewritten:
_
x
 
 = (1/n) å
 		xi = å
 		(1/n) xi
in the latter form the mean of the data set { 2, 7, 4, 9, 4, 3, 6, 3, 7 } is
(1/9) 2 + (1/9) 7 + (1/9) 4 + (1/9) 9 + (1/9) 4 + (1/9) 3 + (1/9) 6 + (1/9) 3 + (1/9) 7
which we can rewrite as
(1/9) 2 + (2/9) 3 + (2/9) 4 + (1/9) 6 + (2/9) 7 + (1/9) 9.
Instead of multiplying each datum by (1/n), we multipy each value taken by the data set by the fraction of the time it occurs. In summation notation this formula is
_
x
 
 = å
 		pixi
where the xi are the different values taken by the data set rather than different points in the data set, pi ³ 0, and åpi = 1. In this notation the variance is s2 = åpi(xi - [`x])2.

0.2  Histograms and summary statistics

The construction of histograms entails grouping data together into classes for better visual presentation. This grouping loses some of the original information, specifically the values of data are replaced by ranges within which the values lie. It is not possible to find the mean or median of the data which provided a histogram, but a ``best'' estimate for the mean or median can be calculated, and bounds on where the mean or median can be are obtainable. The ``best'' estimates are obtained by assuming that the data is uniformly spread within each class.

Example: Consider a histogram which has 10 data in the class with class mark 100, 12 data in the class with class mark 125, 20 data in the class with class mark 150, 8 data in the class with class mark 175, and 5 data in the class with class mark 200. What can you say about the mean and median of the data?

The ``best'' estimate for the mean is obtained by assuming the data is uniformly spread within each interval; for purposes of calculating the mean, this is equivalent to assuming that all the data lie on the class marks. In this example the ``best'' estimate for the mean is

m = 10 ×100 + 12 ×125 + 20 ×150 + 8 ×175 + 5 ×200
55
 = 143 7
11
 .
In order to get bounds on the mean, it is necessary to know the class boundaries, which are halfway between the class marks. Adding or subtracting (25/2) = 12.5 from the class marks provides the class boundaries 87.5, 112.5, 137.5, etc. The least possible mean would occur if all of the data in each class were at the lower class boundary. In this example the least possible mean is
m = (10 ×87.5 + 12 ×112.5 + 20 ×137.5 + 8 ×162.5 + 5 ×187.5)/55 = 131+(3/22).
Similarly, the greatest possible value for the mean is 156+(3/22).

The median is the middle value; uniformly spread data will provide that the area of the histogram on each side of the median will be equal. The total area of this histogram is 10 ×25 + 12 ×25 + 20 ×25 + 8 ×25 + 5 ×25 = 55 ×25 = 1375. Of this sum, 250 comes from the first class, 300 comes from the second class, hence (1375/2) - 550 = 137.5 is needed from the third class to account for half the area. The area 137.5 is obtained from the third class by going (137.5/20) = 6.875 into it. The ``best'' estimate for the median is 144.375, which provides equal area in the histogram on either side of it. Since there are 55 data, the median is the value of the 28th in rank order. This datum will lie in the third class, which contains the 23rd through 42nd data. It is possible that all (most) of the data in the third class would be at the bottom or top of that class, hence the actual value of the median can be anywhere in the range 137.5 to 162.5.

0.3  Other statistics

There are many other statistics which are used. Two which denote relative position which are often encountered are the rank and the z-score. The rank is just what you are familiar with as class rank: it gives the position among the other data, but no raw score. The z-score measures how far a datum is from the mean in terms of standard deviation units. Specifically, zi = [(xi - [`x])/( s)] . The z-score is a measure of relative position.

Example: If a set of data has mean [`x] = 7 and standard deviation s = 4, what is the z-score corresponding to x = 5? The z-score is (5-7)/4 = -.5.

0.4  Exercises

  1. The weights of students in a class are 130, 154, 210, 190, 200, 106, 180, 160, 125, 185, 210, 128, 150, 120, 150, 120, 140, 195, 235, 180, 148, 115, 155, 190, 126, 125, 125, 170, 140, 110, 148, 185, 120, 230, 190, 170, 135, 175, 168, 140, 202, 190, 120, 125, 167, 134, 200, 117, 160, 170, 180, 170, 190, and 120 pounds. Display this information in a histogram. Why did you choose your class marks? Why did you label it the way you did?
  2. The heights of students in a class are 67, 68, 70, 72, 75.5, 63, 69, 71, 63, 70, 73, 69, 70.5, 64, 73, 63, 68, 77, 74, 71, 66.5, 67, 74, 76, 67, 63, 67, 68, 69, 66, 63, 74, 67, 69, 74, 74, 64, 72, 72, 69, 70, 73, 62, 64, 69, 68, 73.5, 64, 72, 73, 72, 71, 75, 66, 64.5, and 68 inches. Display this information in a histogram. Why did you choose your class marks? Why did you label it the way you did?
  3. Give the mean, median, standard deviation, first quartile, third quartile, and inter-quartile range for the weights in problem 1.
  4. Give the mean, median, standard deviation, first quartile, third quartile, and inter-quartile range for the heights in problem 2.
  5. If 10% of the marbles in a jar weigh .4 ounces each, 30% weigh .5 ounces each, 40% weigh .6 ounces each, and 20% weigh .7 ounces each; what are the mean and median weight? What are the standard deviation and inter-quartile range?
  6. If a class has several people between 60 and 78 inches tall, and one midget who is 36 inches tall, which average (mean, median, or midrange) will have the lowest value? Which average will have the highest value?
  7. Why would you expect the mean income of students in your class to be greater than the median? Why would you expect the mean age of students in your class to be higher than the median?
  8. Consider a histogram with 2 data in the class 7.5 - 12.5, 5 data in the interval 12.5 - 17.5, and 4 data in the class 17.5 - 22.5. What is the best estimate for the mean, and what do you know for certain about the mean? What is the best estimate for the median, and what do you know for certain about the median?
  9. For the data in problem 1, what are the z-scores corresponding to the weights 130, 154, and 210 pounds?
  10. For the data in problem 2, what are the z-scores corresponding to the heights 60, 72, and 76 inches?

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