STA220H5 Chapter Notes - Chapter 2.5: Quartile, Box Plot, Interquartile Range
2.5-2.7
For any set of n measurements (arranged in ascending or descending order), the pth
percentile is a number such that p% of the measurements fall below that number
and (100-p)% fall above it.
The lower quartile (QL) is the 25th percentile of a data set. The middle quartile (M) is
the median or 50th percentile. The upper quartile (QU) is the 75th percentile.
The sample z-score for a measurement x is
The population z-score for a measurement x is
Interpretations of z-scores for mound shaped distributions of data
1. Approximately 68% of the measurements will have a z score between -1 and
1.
2. Approximately 95% of the measurements will have a z score between -2 and
2.
3. Approximately 99.7% (almost all) of the measurements will have a z-score
between -3 and 3.
An observation (or measurement) that is unusually large or small relative to the
other values in a data set is called an outlier. Outliers are typically attributable to
one of the following causes:
1. The measurement is observed, recorded, or entered into the computer
incorrectly.
2. The measurement comes from a different population.
3. The measurement is correct but represents a rare (chance) event.
The interquartile range (IQR) is the distance between the lower and upper quartiles:
IQR = QU – QL
Elements of a box plot
1. A rectangle (the box) is drawn with the ends (the hinges) drawn at the lower
and upper quartiles (QL and QU). The median (M) of the data is shown in the
box, usually by a line or a symbol such as +.
2. The points at distances 1.5 (IQR) from each hinge mark the inner fences of
the data set. Lines (the whiskers) are drawn from each hinge to the most
extreme measurement inside the inner fence. Thus,
Lower inner fence = QL – 1.5 (IQR)
Upper inner fence = QU + 1.5 (IQR)
3. A second pair of fences, the outer fences, appears at a distance of 3(IQR) from
the hinges. One symbol e.g.,* is used to represent measurements falling
between the inner and outer fences, and another e.g.,0 is used to
represent measurements that lie beyond the outer fences. Thus, outer fences
are not shown unless one or more measurements lie beyond them. We have
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Document Summary
For any set of n measurements (arranged in ascending or descending order), the pth percentile is a number such that p% of the measurements fall below that number and (100-p)% fall above it. The lower quartile (ql) is the 25th percentile of a data set. The middle quartile (m) is the median or 50th percentile. The upper quartile (qu) is the 75th percentile. The sample z-score for a measurement x is. The population z-score for a measurement x is. An observation (or measurement) that is unusually large or small relative to the other values in a data set is called an outlier. Outliers are typically attributable to one of the following causes: the measurement is observed, recorded, or entered into the computer incorrectly, the measurement comes from a different population, the measurement is correct but represents a rare (chance) event. The interquartile range (iqr) is the distance between the lower and upper quartiles:
