Introduction to Bio statistics

1. Measures of
Central Tendency
DR TUMWEBAZE MATHIAS
BSU-BPH: 2020
25/4/2020 DR MT/BSU/2020 1

2. What are the measures of central tendency?
A measure of central tendency (also referred to as
measures of center or central location) is a summary
measure that attempts to describe a whole set of
data with a single value that represents the middle
or Centre of its distribution
3 – common measures of central tendency:
the mode, the median and the mean.
Each of these measures describes a different
indication of the typical or central value in the
distribution.
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3. Four “Averages”
Mode: The Mode of Variable X is the value of
X that is most frequent.
Median: The Median of X is the value of X
such that an equal number are above and
below X.
Midpoint: Value of X that is half-way between
the smallest and largest value.
Mean: The Arithmetic Average (Sum of X,
divided by the number).
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4. Example: Suppose we have
n = 5 numbers, as follows:
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
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5. Find the Mode:
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
The Mode is the
value of X
that is most frequent.
There are more 10s
than any other value.
So, the Mode is 10.
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6. What is mode :exercise
most commonly occurring value in a distribution
Consider this dataset showing the retirement age of 11
people, in whole years:
54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
1. Present this data in a frequency distribution table. of retirement age
2. what is the mode?
3. What is the median
4. What is the mean.
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7. Limitations of the mode:
1. In some distributions, the mode may not reflect
the Centre of the distribution very well.
When the distribution of retirement age is ordered from
lowest to highest value, it is easy to see that the center of the
distribution is 57 years, but the mode is lower, at 54 years.
54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
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8. Limitations of the mode
2. It is also possible to have more than one mode for the same
distribution of data, (bi-modal, or multi-modal)
3. Presence of more than one mode can limit the
ability of the mode in describing the center or
typical value of the distribution because a single
value to describe the Centre cannot be identified.
4. where the data are continuous, the distribution may
have no mode at all (i.e. if all values are different).
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9. Find the Median
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
The Median has
an equal number
above and below
it. Here, 9 has 2
numbers above
and 2 below, so
the Median is 9.
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10. The median
1. Arrange the observations in increasing or decreasing
order.
2. Find the middle rank with the following formula:
Middle rank = (n +1)
2
3.Identify the value of the median:
a. If the middle rank falls on a specific observation (that is, if
n is odd), the median is equal to the value of that
observation.
b. If the middle rank falls between two observations (that is,
if n is even), the median is equal to the average (i.e., the
arithmetic mean) of the values of those observations.
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11. Median
Find the median of the following set of data with
n = 5: 13, 7, 9, 15, 11
1. Arrange the observations in increasing or decreasing order. We can
arrange them as
either: 7, 9, 11, 13, 15 or: 15, 13, 11, 9, 7
2. Find the middle rank.
Middle rank = (n +1 )
2
(5+1)/2= 6/3 = 3
Therefore, the median lies at the value of the third observation which
is 11
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12. What is the median
The middle value in distribution when the values are
arranged in ascending or descending order.
The median divides the distribution in half (there are
50% of observations on either side of the median
value). In a distribution with an odd number of
observations, the median value is the middle value.
Find the median for the data set below ----------:
54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
When the distribution has an even number of observations,
the median value is the mean of the two middle values.
52, 54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
What is the median in the above data set.DR MT/BSU/2020 12

13. Advantage of the median:
The median is less affected by outliers and skewed data
than the mean, and is usually the preferred measure of
central tendency when the distribution is not symmetrical.
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14. Exercise: Find median of the 5 variablesA-E shown below.
A: 0, 0, 1, 1, 1, 5, 9, 9, 9, 10, 10
B: 0, 4, 4, 4, 5, 5, 5, 6, 6, 6, 10
C: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
D: 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 10
E: 0, 6, 7, 7, 7, 8, 8, 8, 9, 9, 10
1. Arrange the observations in increasing order (already done).
2. Find the middle rank: (11 observations + 1)/2 = 12/2 = 6
3. Identify the value of the median which is the 6th observation:
Median for variables A, B, and C is ?.
Median for variable D = ?
Median for variable E = ?
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15. Find the Midpoint
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9
XMIN + XMAX
2

0+10
2
 5
The Midpoint is half-way
Between the smallest
Value of X and the largest.
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16. The Midrange (Midpoint of an Interval
Midrange is the half-way point or the midpoint of a set of observations.
calculated as the smallest observation plus the largest observation,
divided by two.
Midrange (most types of data) = (x1 + xn )
2
Example
Find the midrange of the 5 variables A-E shown below.
A: 0, 0, 1, 1, 1, 5, 9, 9, 9, 10, 10
B: 2, 4, 4, 4, 5, 5, 5, 6, 6, 6, 9
C: 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11
D: 0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 13
E: 2, 6, 7, 7, 7, 8, 8, 8, 9, 9, 11
1. Rank the observations in order of increasing value (already done)
2. Identify smallest and largest values: for all the distributions
3. Calculate midrange: (x1 + Xn)/2 = - = for each of five distributions
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17. Find the Mean
X1 = 10
X2 = 0
X3 = 10
X4 = 8
X5 = 9

X 
Xi
i1
n

n
4.7
5



Population mean is indicated by the Greek symbol µ .Mean calculated
on a distribution from a sample is indicated by the symbol x̅ (x bar)
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18. What is the mean? Σx/n
The mean is the sum of the value of each
observation in a distribution divided by the
number of observations.
Explaining some terminologies
Σ = (Greek letter sigma) = sum of xi = i-th observation
n or N = the number of observations xi = lowest value in the set of
observations
fi = frequency of xi
xn = highest value in the set of observations
f = total number of observations in interval
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19. Mean
Looking at the retirement age distribution again:
54, 54, 54, 55, 56, 57, 57, 58, 58, 60, 60
The mean is calculated by adding together all the values = --) and dividing by
the number of observations (-- ) which equals ----.
Advantage of the mean:
The mean can be used for both continuous and discrete numeric data.
All values are involved in its calculation's
limitations re involved in its calculation.
- The mean cannot be calculated for categorical data, as the values cannot be
summed.
As the mean includes every value in the distribution in its calculation it is
sensitive to the influence of outliers and skewed distributions
x=56.6 years DR MT/BSU/2020 19

20. Mean and effect of outliers
54+54+54+55+56+57+57+58+58+60+60
what is the mean?----
What happens if the last values 60 is replaced by 81
As the all values are included in the calculation of the mean,
the outlier will influence the mean value.
(54+54+54+55+56+57+57+58+58+60+81 = 644), divided by
11 =
58.5DR MT/BSU/2020 20

21. Symmetrical distributions
When a distribution is symmetrical, the mode, median and mean are all
in the middle of the distribution.
Skewed distributions
In a skewed distribution, the median is often a preferred measure of
central tendency, as the mean is not usually in the middle of the
distribution.
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22. Normal distribution curves
 mode, median and mean are the same and are together in the
centre of the curve
 The important things to note about a normal distribution
are the curve is concentrated in the center and
decreases on either side.
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23. A distribution is said to be positively or right skewed when the tail on
the right side of the distribution is longer than the left side
The data has been grouped into classes, as the variable being measured
(retirement age) is continuous. The mode is 54 years, the modal class is
54-56 years, the median is 56 years and the mean is 57.2 years.
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24. A distribution is said to be negatively or left skewed when the tail
on the left side of the distribution is longer than the right side
The mode is 65 years, the modal class is 63-65 years, the median is 63
years and the mean is 61.8 years. An outlier has a value which is
very different to the rest of the distribution.
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25. symmetrical distribution
positively skewed
negatively skewed
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26. Summary of Example
Case, i Xi
1 10
2 0
3 10
4 8
5 9
Mode= 10
Median = 9
Midpoint = 5
Mean = 7.4
All are “averages”
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