1.3: Descriptive Statistics

• Page ID
259
• Skills to Develop

• Define "descriptive statistics"
• Distinguish between descriptive statistics and inferential statistics

Descriptive statistics are numbers that are used to summarize and describe data. The word "data" refers to the information that has been collected from an experiment, a survey, a historical record, etc. (By the way, "data" is plural. One piece of information is called a "datum.") If we are analyzing birth certificates, for example, a descriptive statistic might be the percentage of certificates issued in New York State, or the average age of the mother. Any other number we choose to compute also counts as a descriptive statistic for the data from which the statistic is computed. Several descriptive statistics are often used at one time, to give a full picture of the data.

Descriptive statistics are just descriptive. They do not involve generalizing beyond the data at hand. Generalizing from our data to another set of cases is the business of  inferential statistics, which you'll be studying in another Section. Here we focus on (mere) descriptive statistics. Some descriptive statistics are shown in Table $$\PageIndex{1}$$. The table shows the average salaries for various occupations in the United States in $$1999$$. (Click  here to see how much individuals with other occupations earn.)

Table $$\PageIndex{1}$$: Average salaries for various occupations in $$1999$$.
Salary Occupation
$112,760 pediatricians$106,130 dentists
$100,090 podiatrists$ 76,140 physicists
$53,410 architects$ 49,720 school, clinical, and counseling psychologists
$47,910 flight attendants$ 39,560 elementary school teachers
$38,710 police officers$ 18,980 floral designers

Descriptive statistics like these offer insight into American society. It is interesting to note, for example, that we pay the people who educate our children and who protect our citizens a great deal less than we pay people who take care of our feet or our teeth.

For more descriptive statistics, consider Table $$\PageIndex{2}$$ which shows the number of unmarried men per $$100$$ unmarried women in U.S. Metro Areas in $$1990$$. From this table we see that men outnumber women most in Jacksonville, NC, and women outnumber men most in Sarasota, FL. You can see that descriptive statistics can be useful if we are looking for an opposite-sex partner! (These data come from the Information Please Almanac.)

Table $$\PageIndex{2}$$: Number of unmarried men per $$100$$ unmarried women in U.S. Metro Areas in $$1990$$.
Cities with mostly men Men per 100 Women Cities with mostly women Men per 100 Women
1. Jacksonville, NC
224
1. Sarasota, FL
66
2. Killeen-Temple, TX
123
68
3. Fayetteville, NC
118
3. Altoona, PA
69
4. Brazoria, TX
117
4. Springfield, IL
70
5. Lawton, OK
116
5. Jacksonville, TN
70
6. State College, PA
113
70
7. Clarksville-Hopkinsville, TN-KY
113
7. Wheeling, WV
70
112
8. Charleston, WV
71
9. Salinas-Seaside-Monterey, CA
112
9. St. Joseph, MO
71
10. Bryan-College Station, TX
111
10. Lynchburg, VA
71

NOTE: Unmarried includes never-married, widowed, and divorced persons, $$15$$ years or older.

These descriptive statistics may make us ponder why the numbers are so disparate in these cities. One potential explanation, for instance, as to why there are more women in Florida than men may involve the fact that elderly individuals tend to move down to the Sarasota region and that women tend to outlive men. Thus, more women might live in Sarasota than men. However, in the absence of proper data, this is only speculation.

You probably know that descriptive statistics are central to the world of sports. Every sporting event produces numerous statistics such as the shooting percentage of players on a basketball team. For the Olympic marathon (a foot race of $$26.2$$ miles), we possess data that cover more than a century of competition. (The first modern Olympics took place in $$1896$$.) Table $$\PageIndex{3}$$ shows the winning times for both men and women (the latter have only been allowed to compete since $$1984$$).

Table $$\PageIndex{3}$$: Winning Olympic marathon times.
Women
Year Winner Country Time
1984 Joan Benoit USA 2:24:52
1988 Rosa Mota POR 2:25:40
1992 Valentina Yegorova UT 2:32:41
1996 Fatuma Roba ETH 2:26:05
2000 Naoko Takahashi JPN 2:23:14
2004 Mizuki Noguchi JPN 2:26:20
Men
Year Winner Country Time
1896 Spiridon Louis GRE 2:58:50
1900 Michel Theato FRA 2:59:45
1904 Thomas Hicks USA 3:28:53
1906 Billy Sherring CAN 2:51:23
1908 Johnny Hayes USA 2:55:18
1912 Kenneth McArthur S. Afr. 2:36:54
1920 Hannes Kolehmainen FIN 2:32:35
1924 Albin Stenroos FIN 2:41:22
1928 Boughra El Ouafi FRA 2:32:57
1932 Juan Carlos Zabala ARG 2:31:36
1936 Sohn Kee-Chung JPN 2:29:19
1948 Delfo Cabrera ARG 2:34:51
1952 Emil Ztopek CZE 2:23:03
1956 Alain Mimoun FRA 2:25:00
1960 Abebe Bikila ETH 2:15:16
1964 Abebe Bikila ETH 2:12:11
1968 Mamo Wolde ETH 2:20:26
1972 Frank Shorter USA 2:12:19
1976 Waldemar Cierpinski E.Ger 2:09:55
1980 Waldemar Cierpinski E.Ger 2:11:03
1984 Carlos Lopes POR 2:09:21
1988 Gelindo Bordin ITA 2:10:32
1992 Hwang Young-Cho S. Kor 2:13:23
1996 Josia Thugwane S. Afr. 2:12:36
2000 Gezahenge Abera ETH 2:10.10
2004 Stefano Baldini ITA 2:10:55

There are many descriptive statistics that we can compute from the data in the table. To gain insight into the improvement in speed over the years, let us divide the men's times into two pieces, namely, the first $$13$$ races (up to $$1952$$) and the second $$13$$ (starting from $$1956$$). The mean winning time for the first $$13$$ races is $$2$$ hours, $$44$$ minutes, and $$22$$ seconds (written $$2:44:22$$). The mean winning time for the second $$13$$ races is $$2:13:18$$. This is quite a difference (over half an hour). Does this prove that the fastest men are running faster? Or is the difference just due to chance, no more than what often emerges from chance differences in performance from year to year? We can't answer this question with descriptive statistics alone. All we can affirm is that the two means are "suggestive."

Examining Table 3 leads to many other questions. We note that Takahashi (the lead female runner in $$2000$$) would have beaten the male runner in $$1956$$ and all male runners in the first $$12$$ marathons. This fact leads us to ask whether the gender gap will close or remain constant. When we look at the times within each gender, we also wonder how much they will decrease (if at all) in the next century of the Olympics. Might we one day witness a sub-$$2$$ hour marathon? The study of statistics can help you make reasonable guesses about the answers to these questions.

Contributors

• Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.

• Mikki Hebl