1.3: Descriptive Statistics
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.)
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 oppositesex partner! (These data come from the Information Please Almanac.)
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. KilleenTemple, TX 
123

2. Bradenton, FL 
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

6. Gadsden, AL 
70

7. ClarksvilleHopkinsville, TNKY 
113

7. Wheeling, WV 
70

8. Anchorage, Alaska 
112

8. Charleston, WV 
71

9. SalinasSeasideMonterey, CA 
112

9. St. Joseph, MO 
71

10. BryanCollege Station, TX 
111

10. Lynchburg, VA 
71

NOTE: Unmarried includes nevermarried, 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 \(1984\)).
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 KeeChung  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 YoungCho  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