Lesson 1.6 · Unit 1 · CED topic 1.7 (part 1)
Measures of center and position
A graph gives you a picture; a summary statistic gives you a number you can quote, compare and calculate with. For one quantitative variable, summary statistics come in three families: measures of center (what is typical), measures of position (where a particular value stands in the distribution), and measures of variability (how spread out the values are). This lesson covers center and position. The next lesson covers variability, the effect of changing units, outliers and the choice of summary.
Learning objective.
- 1.7.A. Calculate measures of center and position for quantitative data.
Essential knowledge.
- 1.7.A.1. Two commonly used measures of center in the distribution of a quantitative variable are the mean and median.
- 1.7.A.2. The mean is the sum of all the values divided by the number of values and can be found with and without using technology. For a sample, the mean is denoted by \(\bar x\): \(\bar x=\dfrac1n\sum_{i=1}^{n}x_i\), where \(x_i\) represents the \(i\)th data point in the sample and \(n\) represents the number of data values in the sample.
- 1.7.A.3. The median is the middle value when the data set is ordered from smallest to largest and can be found with and without using technology. One common method for determining the median of a data set with an even number of values is to use the mean of the two middle values. A common method for determining the median of a data set with an odd number of values is to use the value in the middle of all the values.
- 1.7.A.4. In an ordered data set, the smallest value is the minimum value, and the largest value is the maximum value.
- 1.7.A.5. The first quartile, denoted by Q1, is the median value of the lower half of the ordered data set from the minimum value to the position of the median. Approximately 25% of the values in the data set are less than or equal to Q1. The third quartile, denoted by Q3, is the median value of the upper half of the ordered data set from the position of the median to the maximum value. Approximately 75% of the values in the data set are less than or equal to Q3. The second quartile, Q2, is also the median of the data set. Q1 and Q3 form the boundaries for the middle 50% of values in an ordered data set.
- 1.7.A.6. The pth percentile is the value that has p% of the data less than or equal to it when the data set is ordered from smallest to largest. The first and third quartiles are the 25th and 75th percentiles, respectively.
The mean
The mean is the sum of all the values divided by the number of values. For a sample it is written \(\bar x\) ("x-bar"): \[\bar x=\frac{1}{n}\sum_{i=1}^{n}x_i=\frac{x_1+x_2+\cdots+x_n}{n},\] where \(x_i\) is the \(i\)th data point in the sample and \(n\) is the number of data values. The mean is the balance point of the distribution: if you put the values on a seesaw, it balances at \(\bar x\). It can be found with or without technology.
The median
The median is the middle value when the data are ordered from smallest to largest. If \(n\) is odd, the median is the value in the middle position. If \(n\) is even, a common method is to use the mean of the two middle values. Half of the values are less than or equal to the median and half are greater than or equal to it.
In an ordered data set the smallest value is the minimum and the largest is the maximum.
Quartiles and percentiles
The first quartile, \(Q_1\), is the median of the lower half of the ordered data (from the minimum up to the position of the median). About 25% of the values are less than or equal to \(Q_1\). The third quartile, \(Q_3\), is the median of the upper half (from the position of the median up to the maximum). About 75% of the values are less than or equal to \(Q_3\). The second quartile, \(Q_2\), is the median. \(Q_1\) and \(Q_3\) form the boundaries of the middle 50% of the values.
For an odd number of values, the usual convention (and what the TI-84 does) is to leave the median out of both halves. Different textbooks and software treat the middle value slightly differently, so quartiles can differ a little; AP exam questions are written so that the answer does not depend on the method, and you should always show how you found yours.
The pth percentile is the value that has \(p\%\) of the data less than or equal to it when the data are ordered from smallest to largest. The first and third quartiles are the 25th and 75th percentiles. A percentile describes a position, not a score: a student at the 90th percentile of a test did as well as or better than about 90% of the students, whatever the number of points was.
Find the mean, median, \(Q_1\) and \(Q_3\) for 4, 7, 8, 11, 15.
The mean is \((4+7+8+11+15)/5=45/5=9\). The data are already ordered and \(n=5\) is odd, so the median is the middle (third) value, 8. The lower half (leaving out the median) is 4, 7, so \(Q_1=(4+7)/2=5.5\); the upper half is 11, 15, so \(Q_3=(11+15)/2=13\). The minimum is 4 and the maximum is 15. Notice the mean (9) is a little above the median (8), because the largest value (15) is farther from the middle than the smallest.
Find the mean, median and quartiles of 3, 5, 9, 12, 14, 20.
The mean is \(63/6=10.5\). With \(n=6\) (even), the median is the mean of the two middle values: \((9+12)/2=10.5\). The lower half is 3, 5, 9, so \(Q_1=5\); the upper half is 12, 14, 20, so \(Q_3=14\). Here the mean and median are equal, which happens in a symmetric data set but can also happen by coincidence.
The 30 commute times (minutes) from the graphs lesson have a sum of 781. Find the mean, median, quartiles and the position of a 25-minute commute.
\(\bar x=781/30\approx26.03\) minutes. With \(n=30\) the median is the mean of the 15th and 16th values, 20 and 22, so the median is 21 minutes. The lower half is the first 15 values and its median is the 8th value, \(Q_1=15\). The upper half is the last 15 values and its median is the 23rd value, \(Q_3=32\). (The minimum is 5 and the maximum is 90.) So half of the students commute 21 minutes or less, and the middle 50% commute between 15 and 32 minutes.
For the 25-minute commute: 19 of the 30 values are less than or equal to 25, so it is at about the \(19/30\approx63\)rd percentile; roughly 63% of these students commute 25 minutes or less. The mean (26.03) is larger than the median (21): the long right tail, including the 90-minute commute, pulls the mean up.
Enter the data in L1 (STAT, 1:Edit). Press STAT, arrow to CALC, choose 1:1-Var Stats, enter the list (2nd 1 for L1) and press ENTER. The screen shows \(\bar x\), \(\Sigma x\), \(\Sigma x^2\), \(S_x\) (sample standard deviation), \(\sigma_x\) (population standard deviation), \(n\), then scroll down for minX, \(Q_1\), Med, \(Q_3\) and maxX. Use \(S_x\), not \(\sigma_x\), when the data are a sample.
What the center tells you
The mean and the median measure different things, and both are useful. The mean uses every value, so it reflects the total (a mean of 26 minutes times 30 students is 781 minutes of commuting). The median depends only on the middle of the ordered list, so it is a better description of a typical value when the distribution is skewed or has outliers. Which to report is a judgment you will practice in the next lesson, after you meet the measures of variability.
- \(\bar x=\dfrac{\sum x_i}{n}\) is the mean; the median is the middle of the ordered data (mean of the two middle values if \(n\) is even).
- \(Q_1\) and \(Q_3\) are the medians of the lower and upper halves; the pth percentile has \(p\%\) of the data at or below it.
- Interpret in context: "half of the students commute 21 minutes or less," "25 minutes is about the 63rd percentile."
- On the TI-84, 1-Var Stats gives \(\bar x\), \(S_x\), \(n\), minX, \(Q_1\), Med, \(Q_3\), maxX.
- Find the mean, median, \(Q_1\) and \(Q_3\) of 12, 15, 17, 20, 21, 24 and 29.
Show answer
Mean: \(138/7\approx19.71\). Median: the 4th value, 20. Lower half 12, 15, 17 gives \(Q_1=15\); upper half 21, 24, 29 gives \(Q_3=24\).
- Find the mean, median, \(Q_1\) and \(Q_3\) of 7, 9, 10, 14, 18 and 21.
Show answer
Mean: \(79/6\approx13.17\). Median: \((10+14)/2=12\). Lower half 7, 9, 10 gives \(Q_1=9\); upper half 14, 18, 21 gives \(Q_3=18\).
- A student scored at the 90th percentile on a national reading test. Interpret this in context.
Show answer
The student scored as high as or higher than about 90% of the students who took the test (and about 10% scored higher). It describes position in the group, not the percentage of questions answered correctly.
- A data set has \(n=40\) ordered values. About how many values are less than or equal to \(Q_1\)? To \(Q_3\)?
Show answer
About 25% of 40, or 10 values, are at or below \(Q_1\); about 75% of 40, or 30 values, are at or below \(Q_3\).
- Five numbers have a mean of 12. A sixth number, 18, is added. What is the new mean?
Show answer
The original sum is \(5(12)=60\). The new sum is \(60+18=78\) for 6 numbers, so the new mean is \(78/6=13\).
- The mean salary at a company is \$62,000 and the median is \$48,000. What does this suggest about the distribution of salaries, and why?
Show answer
The mean is much larger than the median, which suggests a distribution skewed right: a few very high salaries pull the mean up while the median stays near the middle of the typical salaries.