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Learn the differences between discrete and continuous variables with examples. Explore stemplots, dotplots, and time-series plots to analyze data effectively.
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Variables A discrete variable is a variable that has gaps between successive, possible values. A continuous variable is a variable that can take on any value between two possible values.
Example: Identifying Discrete and Continuous Variables Identify whether the variable is discrete or continuous. 1. the amount of time (in seconds) it takes a person to run 100 meters 2. the number of times a person has traveled to the Grand Canyon
Solution Identify whether the variable is discrete or continuous. 1. the amount of time (in seconds) it takes a person to run 100 meters The time (in seconds) it takes a person to run 100 meters is a continuous variable because the variable can take on any value between two possible values. For example, not only are 10 seconds and 11 seconds possible times, but the time 10.59329 seconds is also possible.
Solution 2. the number of times a person has traveled to the Grand Canyon The number of times a person has traveled to the Grand Canyon is a discrete variable because there are gaps between successive values. Possible values are 0, 1, 2, 3,…The variable cannot take on values between 0 and 1, between 1 and 2, and so on. For example, a person cannot travel to the Grand Canyon 3.2 times.
Example Construct a stemplot for the test scores we analyzed in Example 2.
Example: Constructing and Interpreting a Dotplot 1. Construct a dotplot of the test scores shown in the table. 2. What observation occurred the most? 3. Each student who scored at most 69 points did not pass the test. How many students did not pass the test? 4. Each student who scored at least 90 points received a grade of A. What proportion of students earned an A on the test? 5. Identify any observations that are quite a bit smaller or larger than the other observations. What are some possible reasons that this happened?
Solution Plotting points on a dotplot is a lot like plotting points on a number line, but we draw the dots above the number line. The first step is to determine that the lowest score is 38 points and the three highest scores are all 100 points. So, we write the numbers 35, 40, 45, …, 100 equally spaced on the number line and write the units “Points” below the numbers. Then for each data value, we draw a dot above the value of the observation, stacking dots if the same value occurs more than once. All the dots should be the same size. (see next slide)
Solution 2. What observation occurred the most? The most number of dots above a number is 4 dots, which are above the test score 85. So, the observation 85 points occurred the most.
Solution 3. Each student who scored at most 69 points did not pass the test. How many students did not pass the test? The phrase “at most 69 points” means less than or equal to 69 points. There are 7 dots to the left of 69 points. So, 7 students did not pass the test.
Solution 4. Each student who scored at least 90 points received a grade of A. What proportion of students earned an A on the test? The phrase “at least 90 points” means 90 or more points. There are 13 dots at 90 points or to the right of 90 points. So, 13 out of 35 students earned As on the test. The proportion is 13/35 about 0.371.
Solution 5. Identify any observations that are quite a bit smaller or larger than the other observations. What are some possible reasons that this happened? The score 38 points is quite a bit lower than the other scores. The student who earned that score might have suffered from math anxiety, have a weak math background, or have skipped studying for the test.
Definitions The frequency of an observation of a numerical variable is the number of times the observation occurs in the group of data. The frequency distribution of a numerical variable is the observations together with their frequencies.
Definitions An outlier is an observation that is quite a bit smaller or larger than the other observations. The kth percentile of some data is a value (not necessarily a data value) that is greater than or equal to approximately k% of the observations and is less than approximately (100 – k)% of the observations.
Example: Percentiles A dotplot of the Test 1 scores is shown. 1. Find the percentile of the test score 70 points. 2. Find the 50th percentile. 3. Suppose the author decides that the test he gave was too difficult. In each past applied calculus class, his cutoff for a C on the first test was approximately at the 15th percentile. He decides to set the cutoff at the 15th percentile. Find its value.
Solution 1. Find the percentile of the test score 70 points. Of the 35 scores, 8 are less than or equal to 70 points. So, the percentage of scores less than or equal to 70 is 8/35 or about 0.23 = 23%. So, the score 70 points is at the 23rd percentile.
Solution 2. Find the 50th percentile. First, we find 50% of 35 scores: 0.50(35) approximately 18 scores. Then we count the dots from left to right until we reach the 18th dot, which is at 85 points. So, 85 points is at the 50th percentile.
Solution 3. Find the cut off for 15th percentile. First, we find 15% of the 35 scores: 0.15(35) about 5 scores. Then we count the dots from left to right until we reach the 5th dot, which is at 65 points So, the cutoff should be 65 points.
Stemplots A stemplot (or stem-and-leaf plot) breaks up each data value into two parts: the leaf, which is the rightmost digit, and the stem, which is the other digits. For the data value 375, the leaf is 5 and the stem is 37.
Example: Constructing a Stemplot Construct a stemplot for the test scores we analyzed in Example 2.
Solution The first data value in Table is 68. The leaf is the rightmost digit, 8, and the stem is the other number, 6. Because the numbers range from 38 to 100, the stems range from 3 to 10, inclusive. We construct a stemplotfor 68 and the otherdata values. The values should be listed least to greatest.
Time-Series Plots To construct a time-series plot, we plot points in a coordinate system where the horizontal axis represents time and the vertical axis represents some other quantity, and we draw line segments to connect each pair of successive dots.
