Statistics often begins with describing a data set using a few useful numbers. Measures of center tell where the data tend to cluster, while measures of spread tell how far the values vary from that center. Both ideas matter because two data sets can have the same average but very different amounts of variability.
A good statistical summary usually includes one measure of center and one measure of spread.
Key Facts
- Mean = sum of all values / number of values.
- Median = middle value when data are ordered from least to greatest.
- Mode = value that appears most often in a data set.
- Range = maximum value - minimum value.
- IQR = Q3 - Q1, where Q1 is the first quartile and Q3 is the third quartile.
- Variance = Σ(x - mean)^2 / n for a population, and standard deviation = sqrt(variance).
Vocabulary
- Mean
- The mean is the arithmetic average found by adding all data values and dividing by the number of values.
- Median
- The median is the middle value of an ordered data set, or the average of the two middle values when there is an even number of values.
- Mode
- The mode is the data value or values that occur most frequently.
- Interquartile Range
- The interquartile range, or IQR, is the spread of the middle 50 percent of the data and equals Q3 minus Q1.
- Standard Deviation
- Standard deviation measures the typical distance of data values from the mean.
Common Mistakes to Avoid
- Reporting only the mean is incomplete because it does not show how spread out the data are.
- Using the range as the only spread measure can be misleading because one extreme value can make the range very large.
- Forgetting to order the data before finding the median or quartiles gives incorrect middle and IQR values.
- Assuming mean, median, and mode are always equal is wrong because skewed or uneven data sets can make these measures different.
Practice Questions
- 1 Find the mean, median, mode, range, and IQR for the data set: 4, 6, 6, 8, 10, 12, 14.
- 2 Two classes have test score means of 80. Class A has a standard deviation of 3, and Class B has a standard deviation of 12. Which class has more consistent scores, and why?
- 3 A data set has most values near 20 but one value at 100. Explain whether the mean or median would better represent the center, and name one spread measure that would help describe the data.