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Ranges
Ranges can also be used to analyze your data. A range (R) lets you summarize numerical observations.
| Range | An index of the extent of the spread of the observations between the empirically determined limits low (l) and high (h). |
The range is determined by subtracting the value of the lowest reported observation from the value of the highest.
Example: Reported income levels show the lowest observation is $20,000 and the highest is $55,000. What is the range?
R = Xh - Xl
R = $55,000 - $20,000
R = $35,000
Quartiles
Quartiles are three values obtained by dividing the sorted dataset into four approximately equal parts. The lower or first quartile is the 25th percentile. The median or second quartile is the 50th percentile. And the upper or third quartile is the 75th percentile. Below is a sample dataset.
| 1.45 | 3.76 | 3.98 | 4.47 | 4.70 | 4.79 | 4.81 |
| 5.25 | 5.40 | 5.97 | 6.58 | 6.85 | 6.88 | 6.90 |
| 7.22 | 7.85 | 7.93 | 7.95 | 8.05 | 8.06 | 8.25 |
| 8.81 | 9.53 | 9.92 | 10.44 | 11.10 | 11.80 | 12.19 |
The quartiles for the above dataset are:
- First quartile: Q1 = (4.81 + 5.25)/2 = 5.03
- Second quartile: Q2 = (6.90 + 7.22)/2 = 7.06
- Third quartile: Q3 = (8.25 + 8.81)/2 = 8.53
Sometimes, a dataset CANNOT be divided into four equal parts (e.g., a dataset of seven). In that case, we interpolate between data items.
Dataset: 1, 2 | 3, 4 | 5| 6, 7
Q1 = (2 + 3)/2 = 2.5
Q2 = the median, 4
Q3 = (5 + 6)/2 = 5.5
Interquartile range
The interquartile range (Q) is most often used as a measure of distribution or spread. Q is equal to the range of the middle 50% of the observations in a distribution. It is equal to the range of the observations that lie between the 25th and 75th percentile, or as previously defined: Q1 (25th Percentile) and Q3 (75th Percentile).
Mathematically, IQR = Q3 - Q1. Using the data above, IQR = 8.53 - 5.03 = 3.5. 3.5 is the range of that half of the data that falls in the middle of the distribution. The interquartile range is more robust than the full range.