Out.Stats3b

What's the average? We have heard the question many times: travelers will ask about on-time averages, university administrators will ask about retention rates (average number of students that return after their first year), investors will ask about average rates of return, sports fans may ask about batting averages,... But how do we go about answering the questions? How do we compute the averages? The first thing to realize is that when people are talking about the average they are talking about a measure of central tendency. In this section we will talk about three measures of central tendency, the mean, median, and mode. To better understand the difference between the three measures, let's return to our example of grades. For an on-line discussion of measures of central tendency you should check out the UCLA On-line Statistics Course, Mathematics 220DX Statistics at the New Hampshire College, DAU the Stat refresher, and Hyperstat Online by David Lane at Rice University.

To demonstrate the measures of central tendency we will use our two grading examples. The frequency distribution of the exam scores for ECN202 is repeated in the first diagram below and the histogram for the grades in ECN201 follows it.

Mean: This is what people generally mean when they say average. The mean is the arithmetic mean of all the data and is defined as the sum of all possible values divided by the number of observations. In the table below we have the grade data for the class that had been divided into four groups. For each group we added up the grades to get a total for the grades and then divided the total by 13, the number of students in each group. If we then add the four group averages and then divide this sum by the number of groups, you get the average of 84.454. This is the same number that you would have obtained if you had simply added up all 52 grades and divided by 52 [Note: this procedure works here only because the groups all had the same number of students. If the groups were of different size, then you would have needed to weight the groups by their relative size.]

In terms of the frequency distribution above, you see that there are not any actual scores of 84 or 85, but those scores are in the middle of the distribution of grades. If there were more grades in the higher range, then this would drag up the mean score.

Student	Grade	Student	Grade	Student	Grade	Student	Grade
1	91.983	14	97.531	27	96.8659	40	89.0001
2	91.7597	15	98.2979	28	76.2859	41	76.5876
3	87.9158	16	70.4242	29	99.6804	42	93.3512
4	77.0586	17	72.6251	30	87.6299	43	88.82
5	98.7479	18	86.9584	31	89.6395	44	77.4919
6	79.8029	19	95.2241	32	85.6969	45	94.7336
7	80.5968	20	91.9544	33	95.2098	46	75.4139
8	77.8953	21	80.2882	34	71.9719	47	86.5368
9	96.1051	22	77.2291	35	92.3448	48	93.7865
10	74.1581	23	93.1482	36	74.4269	49	73.8672
11	83.152	24	75.1727	37	82.7137	50	75.7028
12	91.7678	25	87.5282	38	77.8714	51	73.415
13	78.1368	26	80.501	39	71.828	52	74.7755
Total	1109.08		1106.88		1102.16		1073.48
Group Average	85.3138		85.1448		84.7819		82.5756
Average		84.454

One of the features of this work is the difference between the average for the class and the average for the groups. As we will see in the inferential statistics section, the group averages could be viewed as averages based on a sample of the entire student population. What you see is that the group averages tend to be close to the class average, but they are not exactly the same.

In the ECN201 example to make the computations a bit easier we will examine the descriptive statistics when we take a sample of every fifth student. The sample appears in the table below.

Student	Grade	Student	Grade
1	13	61	22
6	25	66	20
11	19	71	22
16	14	76	24
21	18	81	17
26	27	86	19
31	19	91	26
36	22	96	21
41	25	101	22
46	19	106	22
51	12	111	22
56	25	116	15

The abbreviated descriptive statistics for the sample of ECN201 grades are presented in the table below. [the complete set of statistics can be found on the ECN201grades site]. The "average" grade in the sample is 20.4, fairly close to the mean of 21.7 for the entire population of grades. In the following section we will examine the differences between the average and the other two measures of central tendency.

	Sample	Population
Mean	20.4	21.7
Median	21.5	22
Mode	22	21
Sum	490	2580
Count	24	119

Median: The median is the midpoint in the distribution of grades. At the median score there are as many people above the score as there are below it. The easiest way to do this would be to sort the data by score and pick the midpoint (if there are an even number of observations as there are in this example, you simply average the two middle values). In this ECN202 class, the median grade was 84.42, only slightly below the mean value.

Grade	Student	Grade	Student	Grade	Student	Grade	Student
70.4	1	76.6	14	85.7	27	92.3	40
71.8	2	77.1	15	86.5	28	93.1	41
72.0	3	77.2	16	87.0	29	93.4	42
72.6	4	77.5	17	87.5	30	93.8	43
73.4	5	77.9	18	87.6	31	94.7	44
73.9	6	77.9	19	87.9	32	95.2	45
74.2	7	78.1	20	88.8	33	95.2	46
74.4	8	79.8	21	89.0	34	96.1	47
74.8	9	80.3	22	89.6	35	96.9	48
75.2	10	80.5	23	91.8	36	97.5	49
75.4	11	80.6	24	91.8	37	98.3	50
75.7	12	82.7	25	92.0	38	98.7	51
76.3	13	83.2	26	92.0	39	99.7	52
Median =	(83.2+85.7)		84.4244

The median for the population of ECN201 grades was 22 while the median for the sample was 21.5.

Mode: The mode refers to the most frequently observed number. If we look at the scores and round them to whole numbers and sort them we can find that the number 77 appears four times and the number 92 appears 5 times. These would be the modes in the score distribution.

The mode for the population of ECN201 grades was 21 while the median for the sample was 22.

In this example we have looked at three measures of central tendency, each of which gives us a bit of information on the class' performance on the exam. The fact that they are all different provides the sophisticated observer with additional information. For example, if the mean, median, and mode are the same, you are most likely looking at a symmetric distribution. If the mean is above the median, then you probably have an outlier to the right which would give you a long upper tail that would translate into a skew to the right in the distribution. This is what we would be likely to see in income statistics if there were a few very wealthy people in the sample. If the mean tends to be below the median, then there is a long lower tail - what you would expect to see in a distribution of grades when there was someone with a very low grade. Now it is time to move on to a discussion of variability.

Grades	Grades	Grades	Grades
70	77	86	92
72	77	87	93
72	77	87	93
73	77	88	94
73	78	88	95
74	78	88	95
74	78	89	95
74	80	89	96
75	80	90	97
75	81	92	98
75	81	92	98
76	83	92	99
76	83	92	100