This chapter discusses methods for describing data distributions numerically, including measures of central tendency (mean, median), the five-number summary and box plot, interquartile range, and detecting outliers. It covers calculating the interquartile range and using the 1.5xIQR rule to identify outliers. The chapter also compares standard deviation and interquartile range as measures of data spread, noting that IQR is more robust to outliers and skewed data. Choosing the mean and standard deviation or median and IQR depends on whether the data is symmetric or skewed.

1. STAT 3615: BIOLOGICAL STATISTICS Hamdy F. F. Mahmoud, PhD
Collegiate Assistant Professor
Statistics Department @ VTChapter 2: Describing distributions with numbers – Part II

2. IN THIS CHAPTER, WE COVER
2
 Measuring the center of data by
 Mean
 Median
 5-Number summary and Box plot
 Interquartile range and detecting outliers
 Modified Boxplot in case outliers are existed.
 Measuring spread of data by
 Interquartile range (IQR)
 Standard deviation
 Choosing a good measure for center and for spread.

3. ❑ Interquartile Range (IQR)
IQR is another measure of spread or variability in a data set. The
interquartile range IQR is the distance between the first and the third
quartile, IQR=Q3 - Q1
❑ Detecting outliers:
The 1.5×IQR rule for outliers
➢ Lower limit = Q1 - 1.5×IQR
➢ Upper limit = Q3 + 1.5×IQR
Call an observation an outlier if it is greater than the upper limit or
less than the lower limit.
3
INTERQUARTILE RANGE AND DETECTING OUTLIERS
IQR and outliers

4. 4
INTERQUARTILE RANGE AND DETECTING OUTLIERS
IQR and outliers

5. Example 7: Again spider silk data,
164.0 478.7 251.3 351.7 173.0 448.9 300.6
362.0 272.4 740.2 329.0 327.2 270.5 332.1
288.8 176.1 282.2 236.1 358.2 270.5 290.7
 Calculate the IQR.
 Check if the data have outliers or not.
 Draw the modified boxplot.
5
INTERQUARTILE RANGE AND DETECTING OUTLIERS
IQR and outliers

6.  ANSWER
6
160 200 240 280 320 360 400 440 480 520 560 600 640 680 720 760 800
5-number summary
Min=164
Q1 =260.9
Q2 =290.7
Q3 =354.95
Mix =740.2
➢ IQR
➢ Check for outliers
➢ Modified boxplot

7. Side by side boxplot – more than one boxplot in the same graph
Below is a side-by-side boxplot that shows the average
minutes per day spend in activity (standing and walking )
and in lying down for Lean subjects and Obese subjects.
Conclusion: if you want to be lean, be active!
7
Group
Stand_WalkLie
2121
700
600
500
400
300
200
Data
Boxplot of Lie and Stand_Walk 1 is Lean subjects
2 is Obese subjects• Look at mean
and median
for both
There is a big
difference
between lean and
obese people in
terms of (stand
and walk) time.
There is no much
difference
between lean and
obese people in
terms of lying
down time.

8. SOURCES OF OUTLIERS, AND HOW DO WE DEAL WITH OUTLIERS?
 Human error in recording information
Example 1: in an online survey, undergraduate students
enrolled in a biostatistics course were asked to record their
heights in inches. Out of the 251 numerical values
submitted, two wild observations appeared as 5.3 and 6.
These are obvious errors
Solution: try to correct them.

9. SOURCES OF OUTLIERS, AND HOW DO WE DEAL WITH OUTLIERS?
 Example 2: Perhaps the most famous example of
an outlier caused by data-recording error lies in the
story of Popeye the Sailor. Created in 1929, Popeye is
a friendly cartoon character who attains immediate
superman strength whenever he eats Spanish.
In 1870, a scientific publication reported that
Spanish has iron content more than any other green
leafy vegetable. The iron content of spinach was
miscalculated by a German chemist when he misplaced a
decimal point. While there are just 3.5 milligrams of iron in a
100g serving of spinach, the accepted number became 35
milligrams thanks to his mistake.
Sorry
Popeye,
spinach
DOESN'T
make your
muscles big!

10. SOURCES OF OUTLIERS AND HOW DO WE DEAL WITH OUTLIERS?(CONT.)
 Human error in experimentation or data collection
Example: a researcher records temperature everyday at
12:00pm and one day he recorded it at the evening. In this
case if someone else analyze this data he will find this value
is different from the others.
Solution: remove, but report it.
 Unexplained but apparently legitimate wild observations
Example: most studies in the life sciences are collected by
collecting data about a small sample. Because of this, it can be
difficult to determine whether a suspected outlier in a sample
truly is a wild or just because we study a small sample.
Solution: do the analysis with and without the outliers.

11. Measuring Data Spread or Variability

12. MEASURING SPREAD BY STANDARD DEVIATION
The standard deviation and its close relative, the variance,
measure spread by looking at how far the observations are from
their mean. The variance s2 of a set of observations
is
Or more compactly,
The standard deviation s is the square root of the variance s2,
12
x1, x2,....., xn
s2
=
(x1 -x)2
+(x2 -x)2
+....+(xn -x)2
n-1
s2
=
1
n-1
(xi - x)2
å
s =
1
n-1
(xi - x)2
å

13. A person’s metabolic rate is the rate at which the body
consumes energy. Metabolic rate is important in studies of
weight gain, dieting, and exercise. Here are the metabolic of 7
men who took part in a study of dieting. The units are
kilocalories (Cal) for 24-hour period.
1792 1666 1362 1614 1460 1867 1439
Because men are different in terms of metabolic, we need to
measure how much they are different by calculating variance and
standard deviation.
13
Measuring spread in a data set
Example 8: Metabolic Rate

14. 14
1792
1666
1362
1614
1460
1867
1439
11200
s2
=
1
n-1
(xi - x)2
å
s =
1
n-1
(xi - x)2
å
Measuring spread in a data set
1792-1600= 192
1666-1600=66
-238
14
-140
267
-161
0
1922=36864
4356
56644
196
19600
71289
25921
214870

15. In the variance formula, n-1 (number of observations -1) is
used. That is because number of degree of freedom is n-1!
s measures spread about the mean and should be used only
when the mean is chosen as the measure of center.
s is always greater than or equal zero. s=0 when there is no
spread.
s has the same unit of measurement as the original
observations.
Like the mean, s is not resistant. A few outliers can make s
very large, but less than variance.
15
Measuring spread in a data set
NOTES ON STANDARD DEVIATION

16. MEASURING SPREAD BY INTERQUARTILE RANGE (IQR)
When we have outliers in our data or the distribution is
skewed, the variance and standard deviation are not resistant
(robust) measures for measuring spread. In this case, we
need a resistant measure which is IQR.
IQR = Q3 - Q1
16
Measuring spread in a data set

17. Example 9: Calculate IQR for Metabolic Rate data example
1792 1666 1362 1614 1460 1867 1439

18. 18
In practice
CHOOSING MEASURES FOR CENTER AND SPREAD.
So far, we studied mean and median as measures for
center and variance, standard deviation, and IQR as
measures for spread.
If our data set is symmetric or fairly symmetric, we use
mean and standard deviation. If our data is highly
skewed (right or left) or having outliers, we use median
and IQR to measure center and spread of the data.

19. The End of Chapter 2