Now that we understand the nature of our data, let’s turn to the types of statistics we can use to interpret them. There are 2 types of statistics: descriptive and inferential.
Descriptive 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, an 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.
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 \(\PageIndex{2}\). It 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 |
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, TN-KY |
113 |
7. Wheeling, WV |
70 |
| 8. Anchorage, Alaska |
112 |
8. Charleston, WV |
71 |
| 9. Salinas-SeasideMonterey, 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 \(\PageIndex{3}\): Winning Olympic marathon times.
| Year |
Winner |
Country |
Time |
| Women |
| 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 |
| 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 \(\PageIndex{3}\) 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 far 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.
It is also important to differentiate what we use to describe populations vs what we use to describe samples. A population is described by a parameter; the parameter is the true value of the descriptive in the population, but one that we can never know for sure. For example, the Bureau of Labor Statistics reports that the average hourly wage of chefs is $23.87. However, even if this number was computed using information from every single chef in the United States (making it a parameter), it would quickly become slightly off as one chef retires and a new chef enters the job market. Additionally, as noted above, there is virtually no way to collect data from every single person in a population. In order to understand a variable, we estimate the population parameter using a sample statistic. Here, the term “statistic” refers to the specific number we compute from the data (e.g. the average), not the field of statistics. A sample statistic is an estimate of the true population parameter, and if our sample is representative of the population, then the statistic is considered to be a good estimator of the parameter.
Even the best sample will be somewhat off from the full population, earlier referred to as sampling bias, and as a result, there will always be a tiny discrepancy between the parameter and the statistic we use to estimate it. This difference is known as sampling error, and, as we will see throughout the course, understanding sampling error is the key to understanding statistics. Every observation we make about a variable, be it a full research study or observing an individual’s behavior, is incapable of being completely representative of all possibilities for that variable. Knowing where to draw the line between an unusual observation and a true difference is what statistics is all about.
Inferential Statistics
Descriptive statistics are wonderful at telling us what our data look like. However, what we often want to understand is how our data behave. What variables are related to other variables? Under what conditions will the value of a variable change? Are two groups different from each other, and if so, are people within each group different or similar? These are the questions answered by inferential statistics, and inferential statistics are how we generalize from our sample back up to our population. Units 2 and 3 are all about inferential statistics, the formal analyses and tests we run to make conclusions about our data.
For example, we will learn how to use a t statistic to determine whether people change over time when enrolled in an intervention. We will also use an F statistic to determine if we can predict future values on a variable based on current known values of a variable. There are many types of inferential statistics, each allowing us insight into a different behavior of the data we collect. This course will only touch on a small subset (or a sample) of them, but the principles we learn along the way will make it easier to learn new tests, as most inferential statistics follow the same structure and format.
