9.2: Summarizing Data

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When you get home in the evening and your partner, spouse, roommate, or friend asks you about your day, you usually provide a summary of what happened during the day. This summary gives an impression of the entire day without getting into a lot of the details, unless they are called for. Working with data is very similar. The data observed from a study is usually a large mass of numbers: several variables, hundreds or thousands of observations, all listed out or stored in computer memory somewhere. How can we look at this data in a way that will helps us find the information about research problems that interest us?

To better visualize the issue we face, let us consider a hypothetical study of first-year salaries for those graduating with a bachelor's degree at a particular university. The main issue being considered is whether students of color are thought to have lower first-year salaries after graduation than white students. To study this potential effect, 100 students of color and 100 white students are randomly sampled from the previous graduating class at the university. Each of the sampled students are contacted and the university can observe their first year starting salary. For the sake of simplicity, assume that all the contacted students were employed and reported their salary. The observed salaries, reported in thousands of dollars, are given in Tables 9.1 and 9.2.

Table 9.1 Observed first-year salaries for graduates of color from the university for the hypothetical study discussed in the text (in thousands of dollars).

45	52	50	42	57	49	57	61	52	55
59	53	51	53	58	33	59	64	46	59
53	55	60	59	62	48	48	49	37	49
58	43	53	45	47	42	54	45	38	46
52	53	53	38	53	66	48	37	53	40
47	45	47	54	50	58	41	43	51	59
48	42	43	44	42	48	39	52	45	49
55	61	43	40	45	52	33	45	61	46
37	54	57	57	52	53	43	38	59	53
42	51	57	44	45	45	56	45	47	46

Table 9.2 Observed first-year salaries for white graduates from the university for the hypothetical study discussed in the text (in thousands of dollars).

57	65	56	55	65	64	62	58	68	51
57	57	63	64	46	59	62	45	64	58
62	58	56	49	42	68	59	45	73	54
52	64	67	45	51	72	52	60	62	56
70	49	53	51	59	76	60	63	50	52
59	64	68	64	63	67	69	61	47	57
67	75	64	78	51	58	62	59	55	68
82	62	58	58	49	64	64	60	69	56
56	66	61	45	66	69	56	58	58	65
51	69	55	64	59	63	71	63	64	61

Looking at the tables, what can we learn about the first-year salaries? Briefly, we can observe that in Table 9.1 there seem to be a lot of salaries between $30,000 and $60,000, while in Table 9.2 there seem to be a lot of salaries between $50,000 and $70,000. These observations certainly give one the feeling that in some way the salaries reported in Table 9.1 are generally lower than the salaries reported in Table 9.2, but conceptually the conclusion is somewhat abstract. What would be helpful is a way to summarize the salaries in these tables to make such a comparison easier.

A summary, such as those given at the end of each of the chapters, provides a brief review of the essential points in the information presented. A summary of a table of data will briefly review the essential characteristics of the data. One such summary you are probably familiar with is the computation of an average grade. For Table 9.1 the average income is 49.5, while for Table 9.2 the average income is 60.0. Most people then are very comfortable with the interpreting the result on some intuitive level. The average income of the graduates of color is 49.5 while the average income for white graduates is 60.0, so that the graduates of color, on average, make a little more than $10,000 less than the white graduates in the first year after graduation.

But what have we really done here? What have we really compared? What does an average tell us—what does it summarize about the data in each table? These are important questions. In each case we have taken 100 observed data values and summarized them by a single number. Obviously, much of the detail contained in the data is lost during this calculation and so it is important for us to know what characteristic of the data is being summarized, and what details are left out.

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45	52	50	42	57	49	57	61	52	55
59	53	51	53	58	33	59	64	46	59
53	55	60	59	62	48	48	49	37	49
58	43	53	45	47	42	54	45	38	46
52	53	53	38	53	66	48	37	53	40
47	45	47	54	50	58	41	43	51	59
48	42	43	44	42	48	39	52	45	49
55	61	43	40	45	52	33	45	61	46
37	54	57	57	52	53	43	38	59	53
42	51	57	44	45	45	56	45	47	46

57	65	56	55	65	64	62	58	68	51
57	57	63	64	46	59	62	45	64	58
62	58	56	49	42	68	59	45	73	54
52	64	67	45	51	72	52	60	62	56
70	49	53	51	59	76	60	63	50	52
59	64	68	64	63	67	69	61	47	57
67	75	64	78	51	58	62	59	55	68
82	62	58	58	49	64	64	60	69	56
56	66	61	45	66	69	56	58	58	65
51	69	55	64	59	63	71	63	64	61

45	52	50	42	57	49	57	61	52	55
59	53	51	53	58	33	59	64	46	59
53	55	60	59	62	48	48	49	37	49
58	43	53	45	47	42	54	45	38	46
52	53	53	38	53	66	48	37	53	40
47	45	47	54	50	58	41	43	51	59
48	42	43	44	42	48	39	52	45	49
55	61	43	40	45	52	33	45	61	46
37	54	57	57	52	53	43	38	59	53
42	51	57	44	45	45	56	45	47	46

57	65	56	55	65	64	62	58	68	51
57	57	63	64	46	59	62	45	64	58
62	58	56	49	42	68	59	45	73	54
52	64	67	45	51	72	52	60	62	56
70	49	53	51	59	76	60	63	50	52
59	64	68	64	63	67	69	61	47	57
67	75	64	78	51	58	62	59	55	68
82	62	58	58	49	64	64	60	69	56
56	66	61	45	66	69	56	58	58	65
51	69	55	64	59	63	71	63	64	61

45	52	50	42	57	49	57	61	52	55
59	53	51	53	58	33	59	64	46	59
53	55	60	59	62	48	48	49	37	49
58	43	53	45	47	42	54	45	38	46
52	53	53	38	53	66	48	37	53	40
47	45	47	54	50	58	41	43	51	59
48	42	43	44	42	48	39	52	45	49
55	61	43	40	45	52	33	45	61	46
37	54	57	57	52	53	43	38	59	53
42	51	57	44	45	45	56	45	47	46

57	65	56	55	65	64	62	58	68	51
57	57	63	64	46	59	62	45	64	58
62	58	56	49	42	68	59	45	73	54
52	64	67	45	51	72	52	60	62	56
70	49	53	51	59	76	60	63	50	52
59	64	68	64	63	67	69	61	47	57
67	75	64	78	51	58	62	59	55	68
82	62	58	58	49	64	64	60	69	56
56	66	61	45	66	69	56	58	58	65
51	69	55	64	59	63	71	63	64	61