Learning Outcomes
- Display data graphically and interpret graphs: stemplots, histograms, and box plots
One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit. For example, [latex]23[/latex] has stem two and leaf three. The number [latex]432[/latex] has stem [latex]43[/latex] and leaf two. Likewise, the number [latex]5,432[/latex] has stem [latex]543[/latex] and leaf two. The decimal [latex]9.3[/latex] has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.
Example
For Susan Dean’s spring pre-calculus class, scores for the first exam were as follows (smallest to largest):
[latex]33[/latex]; [latex]42[/latex]; [latex]49[/latex]; [latex]49[/latex]; [latex]53[/latex]; [latex]55[/latex]; [latex]55[/latex]; [latex]61[/latex]; [latex]63[/latex]; [latex]67[/latex]; [latex]68[/latex]; [latex]68[/latex]; [latex]69[/latex]; [latex]69[/latex]; [latex]72[/latex]; [latex]73[/latex]; [latex]74[/latex]; [latex]78[/latex]; [latex]80[/latex]; [latex]83[/latex]; [latex]88[/latex]; [latex]88[/latex]; [latex]88[/latex]; [latex]90[/latex]; [latex]92[/latex]; [latex]94[/latex]; [latex]94[/latex]; [latex]94[/latex]; [latex]94[/latex]; [latex]96[/latex]; [latex]100[/latex]
| Stem | Leaf |
|---|---|
| [latex]3[/latex] | [latex]3[/latex] |
| [latex]4[/latex] | [latex]2[/latex] [latex]9[/latex] [latex]9[/latex] |
| [latex]5[/latex] | [latex]3[/latex] [latex]5[/latex] [latex]5[/latex] |
| [latex]6[/latex] | [latex]1[/latex] [latex]3[/latex] [latex]7[/latex] [latex]8[/latex] [latex]8[/latex] [latex]9[/latex] [latex]9[/latex] |
| [latex]7[/latex] | [latex]2[/latex] [latex]3[/latex] [latex]4[/latex] [latex]8[/latex] |
| [latex]8[/latex] | [latex]0[/latex] [latex]3[/latex] [latex]8[/latex] [latex]8[/latex] [latex]8[/latex] |
| [latex]9[/latex] | [latex]0[/latex] [latex]2[/latex] [latex]4[/latex] [latex]4[/latex] [latex]4[/latex] [latex]4[/latex] [latex]6[/latex] |
| [latex]10[/latex] | [latex]0[/latex] |
The stemplot shows that most scores fell in the [latex]60[/latex]s, [latex]70[/latex]s, [latex]80[/latex]s, and [latex]90[/latex]s. Eight out of the [latex]31[/latex] scores or approximately [latex]26[/latex]% ([latex]\frac{8}{31}[/latex]) were in the [latex]90[/latex]s or [latex]100[/latex], a fairly high number of As.
Try It
For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest):
[latex]32[/latex]; [latex]32[/latex]; [latex]33[/latex]; [latex]34[/latex]; [latex]38[/latex]; [latex]40[/latex]; [latex]42[/latex]; [latex]42[/latex]; [latex]43[/latex]; [latex]44[/latex]; [latex]46[/latex]; [latex]47[/latex]; [latex]47[/latex]; [latex]48[/latex]; [latex]48[/latex]; [latex]48[/latex]; [latex]49[/latex]; [latex]50[/latex]; [latex]50[/latex]; [latex]51[/latex]; [latex]52[/latex]; [latex]52[/latex]; [latex]52[/latex]; [latex]53[/latex]; [latex]54[/latex]; [latex]56[/latex]; [latex]57[/latex]; [latex]57[/latex]; [latex]60[/latex]; [latex]61[/latex]
Construct a stem plot for the data.
The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down [latex]50[/latex] instead of [latex]500[/latex]) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later.
Example
The data are the distances (in kilometers) from a home to local supermarkets. Create a stemplot using the data:
[latex]1.1[/latex]; [latex]1.5[/latex]; [latex]2.3[/latex]; [latex]2.5[/latex]; [latex]2.7[/latex]; [latex]3.2[/latex]; [latex]3.3[/latex]; [latex]3.3[/latex]; [latex]3.5[/latex]; [latex]3.8[/latex]; [latex]4.0[/latex]; [latex]4.2[/latex]; [latex]4.5[/latex]; [latex]4.5[/latex]; [latex]4.7[/latex]; [latex]4.8[/latex]; [latex]5.5[/latex]; [latex]5.6[/latex]; [latex]6.5[/latex]; [latex]6.7[/latex]; [latex]12.3[/latex];
Does the data seem to have any concentration of values?
NOTE
The leaves are to the right of the decimal.
try it
The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers.
[latex]0.5[/latex]; [latex]0.7[/latex]; [latex]1.1[/latex]; [latex]1.2[/latex]; [latex]1.2[/latex]; [latex]1.3[/latex]; [latex]1.3[/latex]; [latex]1.5[/latex]; [latex]1.5[/latex]; [latex]1.7[/latex]; [latex]1.7[/latex]; [latex]1.8[/latex]; [latex]1.9[/latex]; [latex]2.0[/latex]; [latex]2.2[/latex]; [latex]2.5[/latex]; [latex]2.6[/latex]; [latex]2.8[/latex]; [latex]2.8[/latex]; [latex]2.8[/latex]; [latex]3.5[/latex]; [latex]3.8[/latex]; [latex]4.4[/latex]; [latex]4.8[/latex]; [latex]4.9[/latex]; [latex]5.2[/latex]; [latex]5.5[/latex]; [latex]5.7[/latex]; [latex]5.8[/latex]; [latex]8.0[/latex]
Watch this video to see an example of how to create a stem plot.
Example
A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. The two following tables show the ages of U.S. presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.
Presidents’ Ages at Inauguration:
| President | Age | President | Age | President | Age |
|---|---|---|---|---|---|
| Washington | [latex]57[/latex] | Lincoln | [latex]52[/latex] | Hoover | [latex]54[/latex] |
| J. Adams | [latex]61[/latex] | A. Johnson | [latex]56[/latex] | F. Roosevelt | [latex]51[/latex] |
| Jefferson | [latex]57[/latex] | Grant | [latex]46[/latex] | Truman | [latex]60[/latex] |
| Madison | [latex]57[/latex] | Hayes | [latex]54[/latex] | Eisenhower | [latex]62[/latex] |
| Monroe | [latex]58[/latex] | Garfield | [latex]49[/latex] | Kennedy | [latex]43[/latex] |
| J. Q. Adams | [latex]57[/latex] | Arthur | [latex]51[/latex] | L. Johnson | [latex]55[/latex] |
| Jackson | [latex]61[/latex] | Cleveland | [latex]47[/latex] | Nixon | [latex]56[/latex] |
| Van Buren | [latex]54[/latex] | B. Harrison | [latex]55[/latex] | Ford | [latex]61[/latex] |
| W. H. Harrison | [latex]68[/latex] | Cleveland | [latex]55[/latex] | Carter | [latex]52[/latex] |
| Tyler | [latex]51[/latex] | McKinley | [latex]54[/latex] | Reagan | [latex]69[/latex] |
| Polk | [latex]49[/latex] | T. Roosevelt | [latex]42[/latex] | G.H.W. Bush | [latex]64[/latex] |
| Taylor | [latex]64[/latex] | Taft | [latex]51[/latex] | Clinton | [latex]47[/latex] |
| Fillmore | [latex]50[/latex] | Wilson | [latex]56[/latex] | G. W. Bush | [latex]54[/latex] |
| Pierce | [latex]48[/latex] | Harding | [latex]55[/latex] | Obama | [latex]47[/latex] |
| Buchanan | [latex]65[/latex] | Coolidge | [latex]51[/latex] |
Presidents’ Ages at Death:
| President | Age | President | Age | President | Age |
|---|---|---|---|---|---|
| Washington | [latex]67[/latex] | Lincoln | [latex]56[/latex] | Hoover | [latex]90[/latex] |
| J. Adams | [latex]90[/latex] | A. Johnson | [latex]66[/latex] | F. Roosevelt | [latex]63[/latex] |
| Jefferson | [latex]83[/latex] | Grant | [latex]63[/latex] | Truman | [latex]88[/latex] |
| Madison | [latex]85[/latex] | Hayes | [latex]70[/latex] | Eisenhower | [latex]78[/latex] |
| Monroe | [latex]73[/latex] | Garfield | [latex]49[/latex] | Kennedy | [latex]46[/latex] |
| J. Q. Adams | [latex]80[/latex] | Arthur | [latex]56[/latex] | L. Johnson | [latex]64[/latex] |
| Jackson | [latex]78[/latex] | Cleveland | [latex]71[/latex] | Nixon | [latex]81[/latex] |
| Van Buren | [latex]79[/latex] | B. Harrison | [latex]67[/latex] | Ford | [latex]93[/latex] |
| W. H. Harrison | [latex]68[/latex] | Cleveland | [latex]71[/latex] | Reagan | [latex]93[/latex] |
| Tyler | [latex]71[/latex] | McKinley | [latex]58[/latex] | ||
| Polk | [latex]53[/latex] | T. Roosevelt | [latex]60[/latex] | ||
| Taylor | [latex]65[/latex] | Taft | [latex]72[/latex] | ||
| Fillmore | [latex]74[/latex] | Wilson | [latex]67[/latex] | ||
| Pierce | [latex]64[/latex] | Harding | [latex]57[/latex] | ||
| Buchanan | [latex]77[/latex] | Coolidge | [latex]60[/latex] |
Try It
The table shows the number of wins and losses the Atlanta Hawks have had in [latex]42[/latex] seasons. Create a side-by-side stem-and-leaf plot of these wins and losses.
| Losses | Wins | Year | Losses | Wins | Year |
|---|---|---|---|---|---|
| [latex]34[/latex] | [latex]48[/latex] | 1968–1969 | [latex]41[/latex] | [latex]41[/latex] | 1989–1990 |
| [latex]34[/latex] | [latex]48[/latex] | 1969–1970 | [latex]39[/latex] | [latex]43[/latex] | 1990–1991 |
| [latex]46[/latex] | [latex]36[/latex] | 1970–1971 | [latex]44[/latex] | [latex]38[/latex] | 1991–1992 |
| [latex]46[/latex] | [latex]36[/latex] | 1971–1972 | [latex]39[/latex] | [latex]43[/latex] | 1992–1993 |
| [latex]36[/latex] | [latex]46[/latex] | 1972–1973 | [latex]25[/latex] | [latex]57[/latex] | 1993–1994 |
| [latex]47[/latex] | [latex]35[/latex] | 1973–1974 | [latex]40[/latex] | [latex]42[/latex] | 1994–1995 |
| [latex]51[/latex] | [latex]31[/latex] | 1974–1975 | [latex]36[/latex] | [latex]46[/latex] | 1995–1996 |
| [latex]53[/latex] | [latex]29[/latex] | 1975–1976 | [latex]26[/latex] | [latex]56[/latex] | 1996–1997 |
| [latex]51[/latex] | [latex]31[/latex] | 1976–1977 | [latex]32[/latex] | [latex]50[/latex] | 1997–1998 |
| [latex]41[/latex] | [latex]41[/latex] | 1977–1978 | [latex]19[/latex] | [latex]31[/latex] | 1998–1999 |
| [latex]36[/latex] | [latex]46[/latex] | 1978–1979 | [latex]54[/latex] | [latex]28[/latex] | 1999–2000 |
| [latex]32[/latex] | [latex]50[/latex] | 1979–1980 | [latex]57[/latex] | [latex]25[/latex] | 2000–2001 |
| [latex]51[/latex] | [latex]31[/latex] | 1980–1981 | [latex]49[/latex] | [latex]33[/latex] | 2001–2002 |
| [latex]40[/latex] | [latex]42[/latex] | 1981–1982 | [latex]47[/latex] | [latex]35[/latex] | 2002–2003 |
| [latex]39[/latex] | [latex]43[/latex] | 1982–1983 | [latex]54[/latex] | [latex]28[/latex] | 2003–2004 |
| [latex]42[/latex] | [latex]40[/latex] | 1983–1984 | [latex]69[/latex] | [latex]13[/latex] | 2004–2005 |
| [latex]48[/latex] | [latex]34[/latex] | 1984–1985 | [latex]56[/latex] | [latex]26[/latex] | 2005–2006 |
| [latex]32[/latex] | [latex]50[/latex] | 1985–1986 | [latex]52[/latex] | [latex]30[/latex] | 2006–2007 |
| [latex]25[/latex] | [latex]57[/latex] | 1986–1987 | [latex]45[/latex] | [latex]37[/latex] | 2007–2008 |
| [latex]32[/latex] | [latex]50[/latex] | 1987–1988 | [latex]35[/latex] | [latex]47[/latex] | 2008–2009 |
| [latex]30[/latex] | [latex]52[/latex] | 1988–1989 | [latex]29[/latex] | [latex]53[/latex] | 2009–2010 |
