Mean or average

1
Measure of central location --- Contd.

• Mean or average
• Arithmetic mean
• Geometric
• Harmonic
• Median
• Mode
• Mid range
• Mid-hinge
• Quartiles
• Percentiles

Remaining from the last session
2
Measure of central location --- Contd.

• Mid-range Average of the smallest and largest
  observations

• Mid-hinge The average of the first and third
  quartiles.

3
Quartiles Observations that divide data into
four equal parts.

• Box-and-Whisker Plots

4
First Quartile (Q1)
Measure of central location --- Contd.

• Value such that 25 of the ordered observations
  are smaller and 75 of the ordered observations
  are larger

Third Quartile (Q3)

• Value such that 75 of the ordered observations
  are smaller and 25 of the ordered observations
  are larger

5
Measure of other locations -- Contd.
Other Fractiles or Quantiles Quintiles - 5
equal parts Deciles 10 equal
parts Percentiles 100 equal parts

• The fractile relating to the of total observed
  frequency values. For example
• 25th percentile 1st quartile
• 50th percentile Median
• 75th percentile 3rd quartile

6
Measure of other locations -- Contd.

• It is easier to find the value of the required
  fractile item in a grouped frequency distribution
  (The following method is important if you collect
  secondary data for your thesis)
• Find the fractile item by multiplying a fraction
  by the total number of observations
• Example1 the third quartile of students in the
  Biometry class ¾ X 36 27th item
• Example 2 60th percentile of the class would be
  60/10036 21.6 22nd item (round off)

7
Measure of other locations --- Contd.

• Applications
• LC50 lethal concentration (ml/L etc.) of
  certain medicines or chemicals at which 50 of
  the animals die in a certain period of time
• LD50 lethal dose (mg, g/kg animal etc.) at
  which 50 organisms die in a certain period of
  time such as 30 min, 1 hr, etc.
• ED50 effective dose, 50 animals are
  cured/recovered
• In these cases, death of or effect on 50
  organisms is adequate to see the effects, rest
  are not necessary.

8
Measure of other locations --- Contd.

• Percentiles can be used as cut-off values e.g.
  lower than 2.5 and above 97.5 values of the
  distribution will be considered as extreme values
  and thus disregarded. The investigator is
  interested in only the middle 95 of values
  without considering first and last 2.5 values in
  the tail area.

9
Measure of dispersion/variability
Measurement of how scattered or clustered the
data are around the center or central
location Remember - definition of statistics
The scientific study of numerical data based on
variation in nature Science of analyzing data
and drawing conclusions, taking variation into
account
10
Measure of dispersion/variability

• Parameters/statistics of dispersion
• Range
• Quartile range/Quartile deviation
• Mean deviation (MD)
• Standard deviation (SD)
• Variance (Var)
• Standard error (SE)
• Coefficient of variation (CV)

11
Measure of dispersion/variability

• Range Difference between the largest and the
  smallest observations in a set of data
• Range Xlargest Xsmallest
• simplest and rough measure of dispersion
• affected by single extreme/outlying observation
• affected by sample size
• it considers only how far the two extreme values
  are from the center and doesnt take into account
  how and where other observations are clustered or
  dispersed

12
Inter-quartile range/deviation (Mid-spread)
Difference between the Third and the First
Quartiles, therefore, considers data of central
half and ignores the extreme valuesInter-quartil
e Range Q3 - Q1Quartile deviation (Q3 -
Q1)/2
Measure of dispersion/variability
13
Measure of dispersion/variability

• Mean deviation (MD)
• Each observation deviates from the mean
• The average of these deviates is the mean
  deviation or average deviation
• e.g. MD ?(y-?) n
• But in a normal population as 50 observations
  are higher than mean and 50 are lower,
• sum of these deviations is zero (0), therefore,
  later, the absolute difference (ignoring the or
  - signs), MD ?/y-?/ n
• It was popular during early 20th century

14

• Variance (Var)
• Now variance is more popular and widely used
• Has become the basis for analysis, therefore, has
  fundamental importance
• In order to eliminate negative sign, deviations
  are squared (squared units e.g. m2)
• Variance is the average of the squared deviations
• i.e. Variance ?(y-?)2 n

• Standard deviation (SD)
• Positive square root of the variance
• SD v ?(y-?)2 n

15

• Population parameters and sample statistics
• If we are working with samples, the calculation
  under-estimates the variance and SD which is
  biased
• Therefore, instead of using n, n-1 (degrees of
  freedom) is used for sample, e.g.

Population SD
Sample SD
16
Variance and standard deviation are useful for
probability and hypothesis testing, therefore, is
widely used unlike mean deviation
Working formula Variance (S2) ? Xi2
(?Xi)2 /n (n-1) SD (S) v? Xi2 (?Xi)2
/n (n-1)
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Example 1
n 7 Range 2.4-1.21.2g Mean deviation
2.4/7 0.34g s2 ?(y-?) 2/(n-1) 1.12/60.187g2
s v0.187 0.43 g
18
Example 2 Using working formula
n 7 Range 2.4-1.21.2g s2 ?(xi2) (?
xi)2 /n / (n-1) 23.8-(12.6)2/7/6 1.12/6
0.187g2 SD (s) v0.187 0.43 g Coefficient of
Variation (CV) (SD / Mean) 100 (0.43/1.8)
100 24
19

• Standard Error (SE)

• It has become popular recently
• Researchers often misunderstand and mis- use SE
• Variability of observations is SD while
  variability of 2 or more sample means is SE
• Therefore, often called Standard error of the
  means and SD of a set of observations or a
  population

20

• Standard Error (SE)

21
Coefficient of Variation
Measure of dispersion/variability

• Relative measure of variation (data scatter)
  about the mean
• Expressed as a percentage

22

• Understanding the variation

• The more the data is spread out, the larger the
  range, variance, SD and SE (Low precision and
  accuracy)
• The more concentrated the data (precise or
  homogenous), the smaller the range, variance, and
  standard deviation (high precision and accuracy)
• If all the observations are the same, the range,
  variance, and standard deviation 0
• None of these measures can be negative
• Two distant means with little variations are more
  likely to be significantly different and vice
  versa

23
Example there is more chance of having
significant difference in Group A than in B.
Group A Group B
24
Example In which group Trt I and II seem to be
significantly different?
Error bars represent 1SE
60 40 20 0
Group A Group B
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Conclusion Means without variability (SD or
SE) have no meaning SD or SE are used as a
measure of variability which should be presented
along with the means while presenting results
both in tabular and graphical forms. For
examples Table 1. Means for Treatment I and II
( SE).
26
Example 2 Graphical presentation SD or SE are
shown using error bars.
Error Bars
27
See in the Computer Lab after 15 min
Thank you