Mean, Median, and Mode: Simple Statistics Examples
Compare the three common averages and know when each one helps.
Quick Answer
mean = sum / count; even-count median = average of the two middle valuesWhen to Use This
These summaries answer different questions. The mean shares the total equally, the median locates the center of an ordered list, and the mode identifies the most frequent value. Keep all repeated observations: deleting duplicates changes the data.
Step-by-Step Method
- Sort the observations in ascending order and count them.
- Add every observation and divide by the count for the mean.
- For the median, take the middle observation if the count is odd; average the two middle observations if it is even.
- Count frequencies to find the mode or modes.
Calculator Tips and Limits
- The linked basic calculator checks the sum and division for the mean; it does not automatically sort data or find a median/mode.
- Treat missing observations separately. A missing value is not automatically zero.
Worked Examples
Repeated values and an outlier
For 2, 4, 4, 10, the sum is 20 and the count is 4, so the mean is 5. The two middle observations are 4 and 4; their average is 4, the median. The most frequent value is 4, the mode. Adding 100 raises the mean to 120/5 = 24, while the median and mode remain 4.
Two different middle values
For 2, 4, 8, 10, the median is (4 + 8)/2 = 6. Picking either 4 or 8 alone is wrong. Six need not appear in the original data.
Common Mistakes
Choosing just one middle value for an even count
There is no single central observation in an even-length list. Average the central pair.
Ignoring an extreme observation without explanation
An outlier changes the mean, but removing it changes the dataset. Investigate whether it is a valid observation before exclusion.
FAQ
Can a dataset have several modes?
Yes. In 1, 1, 3, 3 both 1 and 3 occur twice and are modes. When all values occur once, elementary descriptions usually report no mode.
What is the mean of an empty list?
It is undefined because the count is zero. Do not report zero as if a measurement had been observed.
References and corrections
NIST: Measures of Location. The worked examples on this page are CalcSolver explanations. Read our calculation and correction method, or report an error with the input and expected result.