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Descriptive Statistics
Author(s)
Mikki Hebl
Prerequisites
none
Learning Objectives
- 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 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 1. Average salaries
for various occupations in 1999.
| $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
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 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
|
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. Clarksville-Hopkinsville, TN-KY |
113
|
7. Wheeling, WV |
70
|
| 8. Anchorage, Alaska |
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.)
The following table shows the winning times for both men and women
(the latter have only been allowed to compete since 1984).
Table 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.
Please answer the questions:
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