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Transcript
```BIOSTATISTICS II
RECAP


ROLE OF BIOSATTISTICS IN PUBLIC
HEALTH
SOURCES AND FUNCTIONS OF VITAL
STATISTICS


RATES/ RATIOS/PROPORTIONS
TYPES OF DATA

CATEGORICAL


NOMINAL /ORDINAL
NUMERICAL

DISCRETE/CONTINOUS/INTERVAL scale/RATIO
VARIABLES


Dependent /
Qualitative /
independent
quantitative
discrete
ordinal
Nominal
dichotomous
continuous
NUMERICAL DATA EXAMINED THROUGH
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Frequency distribution
Percentages, proportions, ratios, rates
Figures
Measures of central tendency
Measures of dispersion
LEARNING OBJECTIVES

From frequency tables to distributions

Types of Distributions: Normal, Skewed

Central Tendency: Mode, Median, Mean

Dispersion: Variance, Standard Deviation
Descriptive statistics are concerned with
describing the characteristics of
frequency distributions
Where is the center?
 What is the range?
 What is the shape [of the
distribution?

Frequency Distributions



Simple depiction of all the data
Graphic — easy to understand
Problems


Not always precisely measured
Not summarized in one number or datum
Frequency Table
Test Scores
Observation
65
70
75
80
85
90
95
Frequency
1
2
3
4
3
2
1
Frequency Distributions
4
3
Frequency
2
1
65
70
75
80
Test Score
85
90
95
Normally Distributed Curve
Skewed Distributions
Characteristics of the Normal
Distribution
It is symmetrical -- Half the cases are to one side of
the center; the other half is on the other side.
The distribution is single peaked, not bimodal or
multi-modal
Most of the cases will fall in the center portion of the
curve and as values of the variable become more
extreme they become less frequent, with “outliers”
at each of the “tails” of the distribution few in
number.
It is only one of many frequency distributions but the
one we will focus on for most of this discussion.
The Mean, Median, and Mode are the same.
Percentage of cases in any range of the curve can be
calculated.
Summarizing Distributions
Two key characteristics of a frequency distribution
are especially important when summarizing data
or when making a prediction from one set of
results to another:
 Central Tendency




What is in the “Middle”?
What is most common?
What would we use to predict?
Dispersion


How Spread out is the distribution?
What Shape is it?
Measures of Central Tendency


The goal of measures of central tendency
is to come up with the one single number
that best describes a distribution of
scores.
Lets us know if the distribution of scores
tends to be composed of high scores or
low scores.
Three measures of central tendency are commonly
used in statistical analysis - the mode, the median,
and the mean
Each measure is designed to represent a typical
score
The choice of which measure to use depends on:
 the shape of the distribution (whether normal or
skewed), and
 the variable’s “level of measurement” (data are
nominal, ordinal or interval).
Appropriate Measures of
Central Tendency

Nominal variables
Mode

Ordinal variables
Median
Interval level variables
Mean
- If the distribution is normal
(median is better with skewed distribution)

Measures of Central Tendency
Mode
The most common observation in a group of scores.

Flavor
f
30
Vanilla
28
25
e
d
ge
Fu
d
R
oc
k
y
R
ip
pl
Ro
a
n
ca
Pe
6
tte
r
Fudge Ripple
Bu
9
ol
ita
n
ry
12
0
ea
p
Butter Pecan
5
N
8
be
r
Neapolitan
10
w
15
St
ra
Strawberry
15
ol
at
e
22
ho
c
Chocolate
20
C

If the data is categorical (measured on the nominal scale)
then only the mode can be calculated.
The most frequently occurring score (mode) is Vanilla.
Va
ni
lla

Distributions can be unimodal, bimodal, or multimodal.
f

Measures of Central Tendency
Mode

The mode can also be calculated with
ordinal and higher data, but it often is not
appropriate.


If other measures can be calculated, the
mode would never be the first choice!
7, 7, 7, 20, 23, 23, 24, 25, 26 has a mode
of 7, but obviously it doesn’t make much
sense.
Median





Middle-most Value
50% of observations are above the
Median, 50% are below it
The difference in magnitude between the
observations does not matter
Therefore, it is not sensitive to outliers
Formula Median = n + 1 / 2
To compute the median
 first you rank order the values of X from low to
high:  85, 94, 94, 96, 96, 96, 96, 97, 97, 98
 then count number of observations = 10.
 divide by 2 to get the middle score  the 5 ½
score
here 96 is the middle score score
Median



Find the Median
4 5 6 6 7 8 9 10 12
Find the Median
5 6 6 7 8 9 10 12
Find the Median
5 6 6 7 8 9 10 100,000
Mean - Average
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
1.
2.

Most common measure of central tendency
Best for making predictions
Applicable under two conditions:
scores are measured at the interval level, and
distribution is more or less normal [symmetrical].
Symbolized as: X
 for the mean of a sample
 μ for the mean of a population
Measures of Central Tendency
Mean



The arithmetic average, computed simply by adding
together all scores and dividing by the number of
scores.
It uses information from every single score.
For a population:
X

X
=
For a Sample: X =
N
n
Finding the Mean


X = (Σ X) / N
If X = {3, 5, 10, 4, 3}
X = (3 + 5 + 10 + 4 + 3) / 5
= 25 / 5
=
5
Find the Mean
Q: 4, 5, 8, 7
A: 6
Median: 6
Q: 4, 5, 8, 1000
A: 254.25
Median: 6.5
IF THE DISTRIBUTION IS
NORMAL
Mean is the best measure of central
tendency
 Most scores “bunched up” in middle
 Extreme scores less frequent 
don’t move mean around.
Measures of Central Tendency ;Mean

If data are perfectly normal, then the mean, median
and mode are exactly the same.

I would prefer to use the mean whenever possible
since it uses information from EVERY score.
Measures of Central Tendency
The Shape of Distributions



With perfectly bell
shaped distributions,
the mean, median, and
mode are identical.
With positively skewed
data, the mode is
lowest, followed by the
median and mean.
With negatively skewed
data, the mean is
lowest, followed by the
median and mode.
Measures of Central Tendency
Using the Mean to Interpret Data
Describing the Population Mean


Remember, we usually want to know
population parameters, but populations are too
large.
So, we use the sample mean to estimate the
population mean.
X 
How well does the mean represent the scores
in a distribution? The logic here is to
determine how much spread is in the
scores. How much do the scores "deviate"
from the mean? Think of the mean as the
true score or as your best guess. If every X
were very close to the Mean, the mean would
be a very good predictor.
If the distribution is very sharply peaked then
the mean is a good measure of central
tendency and if you were to use the mean to
make predictions you would be right or close
much of the time.
Why can’t the mean tell us everything?




Mean describes Central Tendency, what the
average outcome is.
We also want to know something about how
accurate the mean is when making predictions.
The question becomes how good a
representation of the distribution is the mean?
How good is the mean as a description of
central tendency -- or how good is the mean as
a predictor?
Answer -- it depends on the shape of the
distribution. Is the distribution normal or
skewed?
What if scores are widely
distributed?
The mean is still your best measure and your
best predictor, but your predictive power
would be less.
How do we describe this?
 Measures of variability
 Mean Deviation
 Variance
 Standard Deviation
Measures of Variability
Central Tendency doesn’t tell us everything
about how a variable is distributed.
We are most interested in Standard
Deviations (σ) and Variance (σ2)
Dispersion
Once you determine that the variable of interest
is normally distributed, ideally by producing a
histogram of the scores, the next question to be
spread out are the scores around the
mean.
Dispersion is a key concept in statistical
thinking.
The basic question being asked is how much do
the scores deviate around the Mean? The
more “bunched up” around the mean the
better your ability to make accurate
predictions.
Mean Deviation
The key concept for describing normal distributions
and making predictions from them is called
deviation from the mean.
We could just calculate the average distance between
each observation and the mean.
 We must take the absolute value of the distance,
otherwise they would just cancel out to zero!
Formula:
| X  Xi |
 n
Mean Deviation: An Example
Data: X = {6, 10, 5, 4, 9, 8}
X – Xi
Abs. Dev.
1.
7–6
1
7 – 10
3
7–5
2
7–4
3
7–9
2
7–8
1
Total:
X = 42 / 6 = 7
12
2.
3.
4.
Compute X (Average)
Compute X – X and take
the Absolute Value to get
Absolute Deviations
Sum the Absolute
Deviations
Divide the sum of the
absolute deviations by N
12 / 6 = 2
What Does it Mean?

On Average, each observation is two units
away from the mean.
Is it Really that Easy?




No!
Absolute values are difficult to manipulate
algebraically
Absolute values cause enormous problems
for calculus (Discontinuity)
We need something else…
Variance and Standard Deviation
Instead of taking the absolute value, we square
the deviations from the mean. This yields a
positive value.
 This will result in measures we call the Variance
and the Standard Deviation
SamplePopulations: Standard Deviation σ: Standard Deviation
s2: Variance
σ2: Variance

Calculating the Variance and/or
Standard Deviation
Formulae:
Variance:
s 
2
Standard Deviation:
( X  Xi )
N
2
s
(X  X )
i
N
2
Example:
Data: X = {6, 10, 5, 4, 9, 8};
N=6
Mean:
X
X X
(X  X )
6
-1
1
10
3
9
5
-2
4
4
-3
9
9
2
4
Standard Deviation:
8
1
1
s  s 2  4.67  2.16
Total: 42
Total: 28
2
X

X
N
42

7
6
Variance:
s 
2
2
(
X

X
)

N
28

 4.67
6
IN A NORMAL CURVE

AREA CORRESPONDING TO




1 SD WILL COMPRISE 68% OF TOTAL
AREA
2 SD WILL COMPRISE 95% OF TOTAL
AREA
3 SD WILL COMPRISE 99.7% OF TOTAL
AREA
( THE 68- 95-99.7 RULE)
COEFFICIENT OF VARIANCE




set as a proportion of its mean
Expressed as percentage
It is ratio of sample standard deviation to
sample mean. CV of population is based
on expected value and SD of a random
variable
CV = standard deviation/mean x 100
PERCENTILES




Give variability of the distribution
The p’th percentile of distribution is the
value such that p% of observations fall at
or below it
Median is the 50th percentile
Used in calculation of growth charts for
nutritional surveillance and monitoring
QUARTILES


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Values that divide the data into four
groups containing equal numbers of
observations
Quartiles are the 25th and 75th percentiles
First quartile is the median of observations
below the median of the complete data
set,
Third quartile is the median of
observations above the median of the
RANGE

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The range of a sample /data set is the
difference between the largest and
smallest observed value of some
quantifiable characteristic.
A simple summary measure but crude
Like mean it is affected by extreme values
Data: 2,3,4,5,6,6,6,7,7,8,9
RANGE 2- 9= 7
INTERQUARTILE RANGE(IQR)




Calculated by taking difference between
upper and lower quartiles
IQR is the width of an interval which
contains middle 50% of sample
Smaller than range and less affected by
outliers.
Data: 2,3,4,56,6,6,7,7,8,9

Upper quartile=7, lower quartile=4, IQR=3

QUESTIONS ARE WELCOME