What the Mean Tells Us
In Mixed Practice: Describing Distributions in Context, you used numerical summaries alongside graphs to describe distributions. This tutorial focuses on calculating one of those summaries: the sample mean. The mean gives a numerical center for a set of quantitative observations, and it has a useful interpretation as the balance point of the data.
The sample mean is often called the average. To calculate it, add every observation and divide by the number of observations. Each data value contributes to the total, so every observation has equal weight in the calculation. The resulting value is in the same units as the original data.
Here, \(x_1,x_2,\ldots,x_n\) stand for the individual observations, and \(n\) is the number of observations in the sample. The symbol \(\sum x\) means to add all the observed values. Be careful to divide by \(n\), the number of observations—not by the number of distinct values. If the data include repeated values, each occurrence is counted.
The Mean as a Balance Point
Imagine placing the observations as equally weighted points on a number line. The mean is the point where the number line would balance if it could pivot there. Values below the mean pull the balance point to the left; values above it pull to the right. A value farther from the mean has a larger pull, but each observation still counts once.
There is an algebraic way to check this balance-point idea: subtract the mean from each observation, then add those differences. The differences below the mean are negative and those above it are positive. Their sum is zero. In other words, the deviations on one side balance those on the other.
This property is useful for checking a calculation or explaining the mean. It does not mean that the observations themselves must be evenly spaced or symmetric around the mean. Nor does it mean that the mean must be one of the observed values.
A Reliable Calculation Routine
For a short list, use the raw observations. For a frequency table, multiply each value by its frequency before adding. The frequency tells how many times that value occurs, so this method gives each repeated observation its proper contribution.
Read the question carefully. Make sure you know what each value represents and whether you have all the observations in the stated sample.
Count the observations. For a frequency table, add the frequencies to get \(n\).
Add the raw values, or multiply each distinct value by its frequency and add those products.
Divide the total by \(n\), keep the units, and state what the mean represents for the group in the question.
As in Choosing Mean or Median to Describe Center, inspect the distribution and any unusual values before deciding how useful the mean is as a summary. Here the goal is to calculate and interpret the mean itself. Do not treat the word “typical” as a claim that most observations equal the mean; explain it as a summary of the overall center.
Worked Examples: Calculate and Interpret
Worked Example: Time Spent on a Trail
A fictional sample of five hikers records the number of minutes each spent on a short trail: 12, 15, 16, 19, and 23 minutes. Calculate and interpret the sample mean, and check its balance-point property.
Solution—Identify. The observations are five trail times, measured in minutes, so \(n=5\).
Calculate. Add the observations and divide by the number of hikers:
To check the balance point, calculate each observation’s deviation from 17 minutes:
The negative deviations total \(-8\) minutes, and the positive deviations total \(8\) minutes. They balance.
Interpret in context. “For these five fictional hikers, the mean trail time was 17 minutes. This is the balance point of their recorded times and a summary of their average time on the trail.” The mean is also an observed time in this example, but it does not follow that every sample mean must be observed.
Worked Example: Repeated Wait Times in a Frequency Table
A fictional café records the wait, in minutes, for six orders. A frequency table shows that two orders took 4 minutes, three took 5 minutes, and one took 7 minutes. Calculate and interpret the mean wait.
| Wait time (minutes) | Frequency (orders) |
|---|---|
| 4 | 2 |
| 5 | 3 |
| 7 | 1 |
Solution—Find the total and sample size. The sample size is the sum of the frequencies: \(n=2+3+1=6\) orders. To find the total wait time, count each repeated value using its frequency:
Calculate. Divide the total wait by the six orders:
For a balance-point check, the two 4-minute waits are each 1 minute below the mean, while the 7-minute wait is 2 minutes above it. The three 5-minute waits have deviations of zero. Thus, the total deviation is \(2(-1)+1(2)+3(0)=0\) minutes.
Interpret in context. “The mean wait for these six fictional café orders was 5 minutes. It is the balance point and average of the six recorded wait times.” This mean happens to be a recorded value, but the calculation does not say that every order took 5 minutes.
A common shortcut error is to average only the three distinct times, \((4+5+7)/3\). That would give equal weight to each distinct time, not to each order. Multiplying by the frequencies correctly accounts for all six observations.
Worked Example: An Unusually Long Reading Time
A fictional sample records how many minutes six students spent reading a short passage: 8, 9, 10, 10, 11, and 30. Calculate and interpret the sample mean. Explain what the balance-point calculation shows.
Solution—Identify and calculate. There are \(n=6\) reading times, measured in minutes. Their total is \(8+9+10+10+11+30=78\) minutes, so:
The balance-point check uses deviations from 13 minutes:
The five shorter times have a combined deviation of \(-17\) minutes, exactly balancing the \(17\)-minute deviation of the 30-minute observation. This shows how one relatively large value can influence the mean: the balance point must account for that value as well as the other five.
Interpret in context. “For these six fictional students, the mean reading time was 13 minutes. This is the balance point and average of the recorded times.” Here, “average” does not mean that most students took 13 minutes; no student in the sample did. It is still a correct arithmetic mean, but the value 30 minutes pulls the mean upward compared with the five times near 8 to 11 minutes.
As discussed in Why Right Skew Pulls the Mean Above the Median and Choosing Mean or Median to Describe Center, unusual large observations can pull the mean toward larger values. Calculating the mean does not by itself decide which measure best represents a typical observation; that decision also depends on the distribution and the question being asked.
Common Mistakes and AP Exam Tips
- Dividing by the wrong number. Divide by the number of observations, not by the largest value or the number of different values. With a frequency table, add the frequencies to determine \(n\).
- Leaving out repeated observations. A value that appears three times contributes three times to the total. In a frequency table, calculate value times frequency for each row.
- Dropping the units. If the observations are minutes, the mean is in minutes. Include the group, variable, and units in the interpretation.
- Calling the mean the most common value. The mean is an arithmetic average, not a mode. It may not occur among the observations, as the reading-time example shows.
- Claiming that most observations equal the mean. A mean summarizes the overall center. A complete interpretation says what was averaged and for which group, rather than claiming that most individuals had exactly that value.
- Ignoring an unusual value when describing what the mean represents. The calculation still includes every valid observation. If one value is far from the others, explain that it can pull the mean and consider whether the mean is the most useful measure of typical value, as in the earlier tutorial on choosing center.
For full credit, show the sum, the sample size, the division, and a contextual sentence. For example: “The mean time for these five hikers was 17 minutes, found by dividing their total of 85 minutes by 5; 17 minutes is the balance point and average of their recorded trail times.” This states the calculation and interprets the result with the variable, group, and units.
Check Your Understanding
For each question, show the calculation and include an interpretation in context where requested.
- A fictional sample of four bicycle trips has distances of 3, 5, 6, and 10 kilometers. Calculate the mean distance and interpret it as a balance point.
- A frequency table lists temperatures of 18 degrees for 2 mornings, 20 degrees for 3 mornings, and 23 degrees for 1 morning. Find the sample size and mean temperature.
- For the observations 4, 7, and 10, calculate the mean and show that the deviations from the mean add to zero.
- A sample’s mean is 12 seconds, but none of the recorded times is 12 seconds. Is that possible? Explain what the mean represents.
- In a frequency table, a value of 6 occurs four times and a value of 9 occurs once. Explain why averaging only 6 and 9 does not give the mean of the five observations.