Biostatistics Made Ridiculously Simple by Dr. Aryan (Medical Booklet Series by Dr. Aryan Part 8)

Basic Concepts in Biostatistics
Created by:
Anish Dhakal (Aryan)

Preface:
This is the study material designed by Aryan with the sole purpose of revising
and simplifying concepts in biostatistics which many students find difficult and
overwhelming.
Covering everything in one study material is next to impossible. Hence, refer to
gold standard textbooks for building solid concepts or in case of any doubt.
Don’t keep searching for pattern between the consecutive slides. You won’t find
many. Rather to boost your recall and review, I have constructed slides and are
deliberately placed with no much relation between the preceding and the
succeeding ones.
The main rule of a review material is that it must make you recall or learn
maximum amount of information in minimum amount of time and space.
Always remember, everything is literally and absolutely worthless unless you do.
If you know everything in the slides in much detail, you probably wouldn’t need
this material.
Best of luck WORK & SUCCESS! Anish Dhakal (Aryan)

21 Concepts to Master:
1. Normal distribution
2. Skewness
3. Kurtosis
4. Sampling distribution of sample means
5. Central limit theorem
6. Z-score
7. Margin of error
8. Minimum sample size
9. Hypothesis testing
10. p value
11. Critical value
12. , Z-scores, Critical region & Area
under the curve
13. Statement of acceptance or rejection
of claims
14. Probability
15. Binomial and Multinomial probability
16. Discrete probability distribution
17. Fundamental of counting rule
18. Poisson distribution
19. Correlation & Regression
20. Line of best fit
21. Coefficient of determination

Normal Distribution
This is the standard normal bell
shaped curve.
Features of distribution
a) Continuous
b) Symmetric
c) Bell-shaped
Mean is Zero (0) and Standard
Deviation is One (1)
Total area: 1.00 or 100%

Chebyshev’s theorem gives the
formula 1-1/k2
That is for any distribution of
data.
In normal distribution, more
data is concentrated. For
example, the theorem states 75%
of data within 2 S.D. In normal
curve, the number is above 95%

Skewness
If data is perfectly
symmetrical, skewness= Zero
Positive skewness: Right side
of the curve is longer or fatter
(Mean>Median>Mode). As you
can see majority of data is on
the right side: Right skewed)
Negative skewness: Left side
of the curve is longer or fatter
(Mean<Median<Mode). As you
can see majority of data is on
left side: Left skewed)

Kurtosis
Kurtosis is nothing
but the tailedness
of the distribution.

Distribution of Sample Means
Samples are vital as we cannot go and measure the data from large
population every time.
Now, let’s take many random samples from one population each of
size “n”.
Calculate mean of all the samples taken from that population.
Now calculate the mean of all the mean of the samples.
That’s exactly what sampling distribution of sample means is all
about.

Distribution of Sample Means (taken
with replacement)
I. Mean will be same as population mean (µ)
II. Standard Deviation of Sample (from mean)
= Standard error of mean
= SD of population/ 𝑁𝑜. 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒𝑠
= σ/ 𝒏

Central Limit Theorem
When sample size “n” increases without limit, the distribution of
sample means approaches the normal distribution.
If original population is not normally distributed, we need n≥30.
If sample size is less than 30, population must be already normally
distributed.

Z-score
(Value-Mean)/S.D
For population:

X− µ
σ
Hence, for sample means:

ഥ𝑿 −µ
σ/ 𝒏

Confidence Interval of 90%, 95% & 99%
Remember the Critical Value Numbers (two-tailed tests):
 90%: 1.65
 95%: 1.96
 99%: 2.58

Margin of Error
Formula for Interval: ഥ𝑿- Z (σ/√𝒏) < µ < ഥ𝑿 + Z (σ/√𝒏)
If population standard deviation (σ) is not known, use t values and S (standard
deviation of sample). In such cases, n-1 would be the degree of freedom. As degree
of freedom increases further, t distribution would approach normal distribution
among many family of curves with discrete degrees of freedom.
Here, Z (σ/√𝑛) is the margin of error or maximum error of estimate
Maximum error of estimate (E) = Z (σ/√𝒏)
= Z
𝒑𝒒
𝒏
(Margin of error while using proportions)

Minimum Sample Size
n=
Z2 pq
E2 (for proportions)
n =
Z2σ2
E2
The same can also be deduced from the previous formula for margin
of error where,
E = Z (σ/√𝒏) or Z
𝒑𝒒
𝒏

Null and Alternate Hypothesis
Null: H0: µ=k
Alternate: H1
I. H1: µ ≠ k (two-tailed test)
II. H1: µ > k (right-tailed test)
III. H1: µ < k (left-tailed test)
Null hypothesis Errors:
1) Reject when H0 true: Type I Error
2) Do not reject when H0 not true: Type II Error
Maximum probability of committing Type I Error (rejection of null hypothesis when it is true) is
equal to the level of significance (). Confidence level = 1-. 1-ß (ß being the Type II Error) is the
power of the test.

p value
p value denotes how much of your result can occur due to chance
(sampling error).
If the p value is less than the level of significance (), the
hypothesis test is statistically significant
In other words, if your p value is less than the , then your
confidence interval will not contain the null hypothesis value. Hence
you can safely reject the null hypothesis (value in critical or rejection
region).

Confusion Corner: , Z-scores, Critical Region
& Area Under the Curve
 The first aspect is to consider whether it is left, right or two-tailed
curve we are dealing with. If someone says you that the critical value
for 95% confidence interval is 1.96, it assumes that it is a two tailed
test (hence the notation Z /2 read as zee sub alpha over two).
 When you get  (in this case 0.025 on first point), that would give
the area of the curve that we are concerned about. By convention,
while you search for the nearest area in the body of z-table, it is area
to the left of the point. Look for corresponding z-scores (critical
values).
 Alternatively, if you are given z-scores like in the figure alongside,
trace for the area in the table. The first point at -1.96 gives area
0.0250 (blue shaded area on left) and the point +1.96 gives area
0.9750 (blue shaded area on left + non-shaded area in middle).
 The blue shaded area on the right also have area of 0.0250 (its same
as the left side only to be on positive side). If you need area of the
middle portion, that would be 0.9750 - 0.0250 = 0.9500 (95%
confidence level on a two-tailed test with critical region 2.5% on left
and 2.5% on right). Anish Dhakal (Aryan)

Traditional Z-Test for Hypothesis
Testing
1. State null hypothesis and identify the claim (null or alternate hypothesis)
2. Find the critical value (Z-value) on table with  given (based on whether
it is left-tailed: critical value corresponding to , right-tailed: critical
value corresponding to 1-  or two-tailed: critical value corresponding to
/2 and 1- /2 on left and right side respectively)
3. Compute the test value (Z-value)
4. Compare the critical value(s) and computed value in Step 2 & Step 3.
Make a decision about the location of computed value. Does the
computed value fall in the critical region or not? If it falls in critical
region, reject the null hypothesis. Note that here we are comparing z-
values based on  and based on what we calculate. We cannot compare
the areas as that would be same for both positive and negative z-values.

P-value Test for Hypothesis Testing
1. State the hypothesis and identify the claim
2. Compute the test value (Z-value)
3. Find the p-value. Find value corresponding to z-value on the table
(area). If this is a left-tailed test, that corresponding value is your p-
value. If this is a right tailed test, you need to find the rightmost area
beyond the point of z value so use 1-corresponding value. If this is a
two-tailed test, your final p-value would be either double the
corresponding value or double of (1-corresponding value) depending
on whether your z-value is negative or positive respectively.
4. Now all you need to do is compare your p-value with the level of
significance (). On a 5% level of significance, reject null hypothesis
(the difference is significant) if p-value<0.05. If p-value is greater than
or equal to 0.05, there is not enough evidence to reject the null
hypothesis.

How to state the acceptance and rejection of
claims?
Claim is Ho (Null hypothesis):
A. Reject Ho: There is enough evidence to reject null hypothesis
B. Do not reject Ho: There is not enough evidence to reject null
hypothesis
Claim is H1 (Alternate Hypothesis):
A. Reject Ho: There is enough evidence to support alternate hypothesis
B. Do not reject Ho: There is not enough evidence to support alternate
hypothesis

Concept of Hypothesis Testing
While you test a hypothesis, never simply say that null hypothesis is
true or false. You do not know that!
The only thing you know is that based on evidence provided, there is
enough evidence to reject the null hypothesis or not. To state with
100% certainty whether that is true or false, whole population needs
to be tested.
When a null hypothesis is rejected at a level of significance , the
confidence interval computed at 1- would not contain the value of
mean stated by the null hypothesis and vice versa. That’s pretty
obvious.

Probability
Classical probability: all outcomes equally likely to happen (sample
spaces)
Empirical probability: actual experiments to determine probability
(frequency distribution)
Conditional probability: The probability that second event B occurs
given that the first event A has occurred can be found by:
P(B|A) = P(A and B)/P(A)

Fundamental of Counting Rule
I. If repetitions are permitted, the numbers stay the same going from
left to right. For example if a number of 5 digits is to be selected
the total number of possibilities = 10*10*10*10*10 = 100000
possibilities of selecting a 5 digit number.
II. If repetitions are not permitted, we got one less choice every time.
The numbers decrease by one for each place left to right. Total
number of possibilities in the above example = 10*9*8*7*6 =
30240 possibilities of selecting a 5 digit number.

Permutation and Combination
Permutation of ‘n’ objects taking ‘r’ objects at a time (in specific order):
𝒏 𝑷 𝒓 =
𝒏!
𝒏−𝒓 !
Combination of ‘r’ object selected from ‘n’ objects:
nCr =
𝒏!
𝒏−𝒓 !𝒓!
Hence, nCr = nPr
𝒓!
(r! removes the duplicates which have great
significance in permutation. 1 & 2 or 2 & 1 would be same in combination)

Mean of random variable with discrete
probability distributions:
µ = X1.P(X1) + X2.P(X2) + ………………………………... + Xn.P(Xn) = Σ[X.P(X)]
where,
X1, X2……………….Xn are the outcomes
P(X1), P (X2)………P(Xn) are the corresponding probabilities

Binomial Distribution of Probability
Condition for binomial probability experiment:
i. Fixed number of trials
ii. Two outcomes or results can be reduced to two outcomes
iii. Outcomes of each trial independent of each other
iv. Probability of success remains the same for each trial
P(x=k) = b(k; n,p) =
𝒏!
𝒏−𝒌 !𝒌!
.pk.qn-k = C(n,k).pk.qn-k
where,
p= probability of success
q= probability of failure
n= number of trials
k= number of success (at x=k) (0≤x≤n) Anish Dhakal (Aryan)

Multinomial Distribution
P(x) =
𝒏!
x1!x2!x3!....................xk!
. 𝐩1
x1.p2
x2……….pk
xk
where,
x1,x2,x3…………………..xk are number of occurrence of events
p1, p2, p3………………..pk are the corresponding probabilities
x1+x2+x3………………….xk= n (total number of events)
p1+p2+p3………………….pk = 1

Poisson Distribution
P(x, λ)=
e−ʎ.ʎx
𝒙!
where,
ʎ= mean number of occurrence per unit time, length, area or volume
x= number of occurrence of the event
e= 2.7183
n: sufficiently large
probability of success: sufficiently small

Correlation Vs. Regression
Correlation:
simply determines whether two variables are correlated and to what extent.
Regression:
determines nature of relationships, estimate dependent variable based on
independent variable (functional relationship/projection of events).

Line of Best Fit
Choose a straight line which best represents the scatter plot and you
have got the line of best fit.
Sum of squares from each point to the line is minimum.
Equation of the line (Regression line equation):
Predicted value (y’)= a+bx
where,
a= y-intercept
x= slope of line
Closer the observed value (y) is to the predicted value (y’), the better is the
fit and the closer ‘r’ is to +1 or -1.

Total variation = Explained variation + Unexplained variation
= Σ(y’-തy)2 + Σ(y-y’)2
= Σ(y-തy)2
y-y’ is the unexplained deviation or residuals. Sum of the square of residuals Σ(y-y’)2 being
the least possible value gives rise to line of best fit.

Coefficient of determination
r2 =
explained variation
total variation
=
Σ(y’−ഥy)2
Σ(y−ഥy)2
This is the percentage of total variation explained by the regression
line using the independent variable.
1-r2: coefficient of non-determination: due to chance

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Biostatistics Made Ridiculously Simple by Dr. Aryan (Medical Booklet Series by Dr. Aryan Part 8)

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Biostatistics Made Ridiculously Simple by Dr. Aryan (Medical Booklet Series by Dr. Aryan Part 8)