1. Measure of the Central Tendency For Grouped data

2. Mean – Grouped Data
The mean may often be confused with the median, mode or range. The mean is the arithmetic average of a set of values, or distribution.
Example: The following table gives the frequency distribution of the number
of orders received each day during the past 50 days at the office of a mail-order
company. Calculate the mean.
Number
of order f
10 – 12
13 – 15
16 – 18
19 – 21 4 12 20 14
n = 50
Solution:
X is the midpoint of the class. It is adding the class limits and divide by 2.
Number
of order f x fx
10 – 12
13 – 15
16 – 18
19 – 21 4 12 20 14 11 17 44
168
340
280
n = 50
= 832

3. Median and Interquartile Range – Grouped Data
a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution,
Step 1: Construct the cumulative frequency distribution.
Step 2: Decide the class that contain the median.
Class Median is the first class with the value of cumulative
frequency equal at least n/2.
Step 3: Find the median by using the following formula:
Where:
n = the total frequency
F = the cumulative frequency before class median
= the frequency of the class median
i = the class width
= the lower boundary of the class median

4. Example: Based on the grouped data below, find the median:
Time to travel to work
Frequency
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50 8 14 12 9 7
Solution:
1st Step: Construct the cumulative frequency distribution
Time to travel
to work
Frequency
Cumulative Frequency
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50 8 14 12 9 7 22 34 43 50
class median is the 3rd class
So, F = 22, = 12, = 20.5 and i = 10

5. Therefore,
Thus, 25 persons take less than 24 minutes to travel to work and another 25 persons
take more than 24 minutes to travel to work.

6. Mode – Grouped Data Mode
Mode is the value that has the highest frequency in a data set.
For grouped data, class mode (or, modal class) is the class with the highest frequency.
To find mode for grouped data, use the following formula:
Where:
i is the class width
is the difference between the frequency of class mode and the frequency
of the class after the class mode
is the difference between the frequency of class mode
and the frequency of the class before the class mode
is the lower boundary of class mode

7. Calculation of Grouped Data - Mode
Example: Based on the grouped data below, find the mode
Time to travel to work
Frequency
1 – 10
11 – 20
21 – 30
31 – 40
41 – 50 8 14 12 9 7
Solution:
Based on the table,
= 10.5, = (14 – 8) = 6, = (14 – 12) = 2 and
i = 10

8. Lecture 4 Descriptive Statistics “Indicators of dispersion”
Biostatistics
Lecture 4
Descriptive Statistics
“Indicators of dispersion”
Dr. Alkilany 2012

9. Indicators of dispersion
If all observations are the same, there is no variability. If they are not all the same, then dispersion is present in the data.
Variation is an inherent characteristic of experimental observations due to several reasons.
it is always important to get an estimate of how much given objects tend to differ from that central tendency
In any experiment, variation will depend on:
The instrument used for analysis.
The analyst performing the assay.
The particular sample chosen.
Unidentified error commonly known as noise.
Dispersion (variability)
Central tendency
Dr. Alkilany 2012

10. Indicators of dispersion (why we need them?)
1. A, B, C have the same mean
Based on similarity of the mean, can we say the data sets are the same?
What is the differences between these data sets?
How we can describe the (differences)? A B C
Dr. Alkilany 2012
Dr. Alkilany 2012

11. Indicators of dispersion
Standard deviation
Coefficient of variation
Variance
Quartiles
Box and whisker plot
Dr. Alkilany 2012

12. Standard deviation Alpha Bravo Mean~250 mg/tablet Mean~250 mg/tablet
Two tabletting machines producing erythromycin
tablets with a nominal content of 250 mg.
500 tablets are randomly selected from each machine
and their erythromycin contents was assayed.
Mean~250 mg/tablet
Mean~250 mg/tablet
Although tablets from both machines had equal mean, do you
think the two machine still differ? How?
Dr. Alkilany 2012

13. Standard deviation
The two machines are very similar in terms of average drug content for the tablets, both producing tablets with a mean very close to 250 mg. However, the two products clearly differ.
With the Alpha machine, there is a considerable proportion of tablets with a content differing by more than 20 mg from the nominal dose (i.e. below 230 mg or above 270 mg), whereas with the Bravo machine, such outliers are much rarer.
An ‘indicator of dispersion’ is required in order to convey this difference in variability and to decide which one has better performance!!
Dr. Alkilany 2012

14. Let us go back to tabletting machines (raw data)!
Standard deviation
This is the standard deviation (SD) for the sample
For population it is usually donated : σ
Same unit of the mean
Standard deviation is a widely used measure of variability and central dispersion
Let us go back to tabletting machines (raw data)!
Dr. Alkilany 2012

15. Standard deviation Xi __ X=
Dr. Alkilany 2012

16. Standard deviation
The Alpha machine produces rather variable tablets and so several of the tablets deviate considerably from the overall mean.
These relatively large figures then feed through the rest of the calculation, producing a high final SD (8.72 mg).
In contrast, the Bravo machine is more consistent and individual tablets never have a drug content much above or below the overall average.
The small figures in the column of individual deviations, leading to a lower SD (3.78 mg).
Dr. Alkilany 2012

17. Standard deviation Reporting the SD:
The  symbol is used in reporting the SD
The symbol  reasonably interpreted as meaning ‘more or less’.
 is used to indicate variability.
With the tablets from our two machines, we would report their drug contents as:
Alpha machine: 248.78.72 mg (MeanSD mg)
Bravo machine: 251.13.78 mg (MeanSD mg)
The figures quoted before summarize the true situation. The two machines produce tablets with almost identical mean contents, but those from the Alpha machine are two to three times more variable.
Dr. Alkilany 2012

18. Standard deviation and Coefficient of variation
- Elephant tail=150±10 cm
- Mouse tail=7±3 cm
With this in mind, which is more variable: the elephant tail length results or the one for the mouse?
Elephant tail: CV=10/150x100=6.7%
Mouse tail: CV= 3/7x100=42.8%
Coefficient of variation (CV) expresses variation relative to the magnitude of data
Useful to compare variation in two or more sets of data with different mean values
CV is has no unit (it is a ratio!)
Dr. Alkilany 2012

19. Variance
The Variance: 2 (population) or S2 (sample) is a measure of spread that is related to the deviations of the data values from their mean.
Unit: same as mean but squared. If mean in mg, variance will be in mg2
sample
Population
Dr. Alkilany 2012

20. Quartiles Q2
Median Q1 Q3
Quartiles: the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled
The median= Q2
First Quartile Q1
cuts off lowest 25% of data
25th percentile
Second Quartile Q2
cuts data set in half
50th percentile
Third Quartile Q3
cuts off highest 25% of data, or lowest 75%
75th percentile
Dr. Alkilany 2012

21. Interquartile range: difference between the upper and lower quartiles
IQR= (Q3 – Q1)
Dr. Alkilany 2012

22. Finding Quartiles
To find the quartiles for a set of data, do the following:
Arrange the data from smallest to highest (ordered array)
Locate the median (Q2)
The half to the left: locate their median (Q1)
The half to the right: Locate their median (Q3)
Half to the left
Half to the right
Median Q1 Q2 Q3
Dr. Alkilany 2012

23. Finding Quartiles Example with odd (n)
Times needed for 15 tablets to disintegrate in minutes: 5,
Data is already in an order from smallest to highest
Median is the (n+1/2)th=8th=20 (in bold red)
For the half to the right: n=7, median=4th=10 minutes
For the half to the right: n=7, median=4th=30 minutes
Q1=10 minutes; Q2= 20 minutes; Q3=30 minutes. IQR=Q3-Q1=20 minutes
This means that 25% of tablets need less than 10 minutes to disintegrate. Also 50% of tablets need 20 minutes to disintegrate. Before 30 minutes, 75% of all tables were disintegrated. 25% only of these tablets need more than 30 minutes to disintegrate.
This question can come in this form
Disintegration time (min)
Frequency 5 1 10 4 12 15 20 25 2 30 40 60
Total
Dr. Alkilany 2012

24. Finding Quartiles Example with even (n)
Times needed for 20 capsules to disintegrate in minutes:
5, 10, 10, 15, 15, 15, 15, 20, 20, 20, 25, 30, 30 40, 40, 45, 60, 60, 65, 85
Data is already in an order from smallest to highest
Median is the mean of the two middle values (n/2)th and ((n/2) + 1)th (in bold red)=10th and 11th=(20+25)/2= 22.5
For the half to the right: n=10, median=mean of 5th & 6th=15 minutes
For the half to the right: n=10, median=mean of 5th & 6th=42.5
Q1=15minutes; Q2= 22.5 minutes; Q3=42.5 minutes. IRQ=??
Disintegration time (min)
Frequency 5 1 10 2 15 4 20 3 25 30 40 45 60 65 85
Total
Dr. Alkilany 2012

25. Quartiles
Consider the elimination half-lives of two synthetic steroids have been determined using two groups, each containing 15 volunteers.
The results are shown in the following table with the values ranked from lowest to highest for each steroid.
Dr. Alkilany 2012

26. Quartiles and IQR as a measurement for data spread
The IQR for the half life of steroid 2 is only half that for steroid 1, duly reflecting its less variable nature.
Just as the median is a robust indicator of central tendency, the interquartile range is a robust indicator of dispersion. The interquartile range is a more useful measure of spread than range as it describes the middle 50% of the data values and thus less affected by outliers. 

27. Box and whisker plot
A box-and-whisker plot can be useful for handling many data values.
It shows only certain statistics rather than all the data.
Five-number summary is another name for the visual representations of the box-and-whisker plot.
The five-number summary consists of the median, the quartiles, and the smallest and greatest values in the distribution (not including outliers).
Immediate visuals of a box-and-whisker plot are the center, the spread, and the overall range of distribution.

28. Box and whisker plot
The first step in constructing a box-and-whisker plot is to first find the median (Q2), the lower quartile (Q1) and the upper quartile (Q3) of a given set of data.
Example: The following set of numbers are weights of 10 patients in hospital (kgs): 75 1 62
Smallest (S) 2 67 78 3 73 Q1 96 4 5
Median=78.5 (Q2) 93 6 79 85 7 81 8 Q3 9 10
Largest (L) S L Q3 Q1 Q2

29. FYIP
Outliers (extreme values) are values that are much bigger or smaller (distant) than the rest of the data.
In order to be an outlier, the data value must be:
larger than Q3 by at least 1.5 times the interquartile range (IQR), or
smaller than Q1 by at least 1.5 times the IQR.
Represented by a dot on the box and whisker plot