Data Science interview questions of Statistics

Statistics for Data Science - Interview Questions
1. What is Inferential Statistics?
Inferential statistics is a statistical method that deduces from a small but
representative sample the characteristics of a bigger population. In other words, it
allows the researcher to make assumptions about a wider group, using a smaller
portion of that group as a guideline.
2. What is the difference between Population and Sample?
From the population we take a sample. We cannot work on the population either
due to computational costs or due to availability of all data points for the population.
From the sample we calculate the statistics. From the sample statistics we conclude
about the population.
3. What is the relationship between mean and median in normal distribution?
In the normal distribution mean is equal to median
4. What is an outlier?
An outlier is an abnormal value (It is at an abnormal distance from the rest of the
data points).
5. What can I do with Outliers?
● Remove outlier
When we know the data-point is wrong (negative age of a person)
When we have lots of data
We should provide two analyses. One with outliers and another without.
● Keep outlier
When there are lot of outliers (skewed data)
When results are critical
When outliers have meaning (fraud data)
6. What is the difference between population parameters and sample statistics?
● Population parameters are:
Mean = µ
Standard deviation = σ
● Sample statistics are:
Mean = x (bar)
Standard deviation = s
7. What is the difference between inferential statistics and descriptive statistics?
● Descriptive statistics – provides exact and accurate information.
● Inferential statistics – provides information of a sample and we need inferential
statistics to reach to a conclusion about the population.
STATISTICS REVISION MATERIAL/INTERVIEW QUESTION AUTHOR: DHRUB SATYAM

8. What is the difference between population and sample in inferential statistics?
● From the population we take a sample. We cannot work on the population either
due to computational costs or due to availability of all data points for the
population.
● From the sample we calculate the statistics
● From the sample statistics we conclude about the population
9. What are descriptive statistics?
● Descriptive statistic is used to describe the data (data properties)
● 5-number summary is the most commonly used descriptive statistics
10. Most common characteristics used in descriptive statistics?
● Central Tendency– middle of the data. Mean / Median / Mode are the most
commonly used as measures.
Mean – average of all the numbers
Median – the number in the middle
Mode – the number that occurs the most. The disadvantage of using Mode is
that there may be more than one mode.
● Spread – How the data is dispersed. Range / IQR / Standard Deviation / Variance
are the most commonly used as measures.
Range = Max – Min
Inter Quartile Range (IQR) = Q3 – Q1
Standard Deviation (σ) = √(∑(x-µ)2 / n)
Variance = σ2
● Shape – the shape of the data can be symmetric or skewed
Symmetric – the part of the distribution that is on the left side of the median is same
as the part of the distribution that is on the right side of the median
Left skewed – the left tail is longer than the right side
Right skewed – the right tail is longer than the left side
● Outlier – An outlier is an abnormal value
Keep the outlier based on judgement
Remove the outlier based on judgement
11. What are 5 Point Summary (or 5 number summary)?
● Low extreme (minimum)
● Lower quartile (Q1)
● Median
● Upper quartile (Q3)
● Upper extreme (maximum)
12. What are the measures of central tendency? When to use which one?
Measures of central Tendency are – Mean, Median & Mode.

a. Mean is the most frequently used measure of central tendency and generally
considered the best measure of it. However, there are some situations where
either median or mode are preferred.
b. Median is the preferred measure of central tendency when:
● There are a few extreme scores in the distribution of the data. (NOTE:
Remember that a single outlier can have a great effect on the mean). b.
● There are some missing or undetermined values in your data. c.
● There is an open ended distribution (For example, if you have a data field
which measures number of children and your options are 0, 1, 2, 3, 4, 5 or
“6 or more,” then the “6 or more field” is open ended and makes
calculating the mean impossible, since we do not know exact values for this
field) d.
● You have data measured on an ordinal scale.
c. Mode is the preferred measure when data are measured in a nominal (and even
sometimes ordinal) scale.
13. What is CLT? What is its significance?
“The central limit theorem states that the sampling distribution of the sample mean
approaches a normal distribution as the sample size gets larger no matter what the
shape of the population distribution.”
The central limit theorem is important because it is used in hypothesis testing and
also to calculate confidence intervals.
(Please follow for more reference:
https://towardsdatascience.com/understanding-the-central-limit-theorem-e598158cc5da)
14. What is p-value?
The P value, or calculated probability, is the probability of finding the observed, or
more extreme, results when the null hypothesis (H0) of a study question is true – the
definition of ‘extreme’ depends on how the hypothesis is being tested. P is also
described in terms of rejecting H0 when it is actually true, however, it is not a direct
probability of this state.
When you perform a hypothesis test in statistics, a p-value helps you determine the
significance of your results. Hypothesis tests are used to test the validity of a claim
that is made about a population. This claim that’s on trial, in essence, is called the
null hypothesis.
The alternative hypothesis is the one you would believe if the null hypothesis is
concluded to be untrue. The evidence in the trial is your data and the statistics that
go along with it. All hypothesis tests ultimately use a p-value to weigh the strength of
the evidence (what the data are telling you about the population). The p-value is a
number between 0 and 1 and interpreted in the following way:

● A small p-value (typically ≤ 0.05) indicates strong evidence against the null
hypothesis, so you reject the null hypothesis.
● A large p-value (> 0.05) indicates weak evidence against the null hypothesis,
so you fail to reject the null hypothesis.
● p-values very close to the cut-off (0.05) are considered to be marginal (could
go either way). Always report the p-value so your readers can draw their own
conclusions.
15. What are z and t stats?
Z-test:
In a z-test, the sample is assumed to be normally distributed. A z-score is calculated
with population parameters such as “population mean” and “population standard
deviation” and is used to validate a hypothesis that the sample drawn belongs to the
same population.
Null: Sample mean is same as the population mean
Alternate: Sample mean is not same as the population mean
The statistics used for this hypothesis testing is called z-statistic, the score for which
is calculated as
● z = (x — μ) / (σ / √n), where
● x= sample mean
● μ = population mean
● σ / √n = population standard deviation
If the test statistic is lower than the critical value, accept the hypothesis or else reject
the hypothesis
T-test:
A t-test is used to compare the mean of two given samples. Like a z-test, a t-test also
assumes a normal distribution of the sample. A t-test is used when the population
parameters (mean and standard deviation) are not known.
There are three versions of t-test
1. Independent samples t-test which compares mean for two groups
2. Paired sample t-test which compares means from the same group at different
times
3. One sample t-test which tests the mean of a single group against a known mean.
The statistic for this hypothesis testing is called t-statistic, the score for which is
calculated as
● t = (x1 — x2) / (σ / √n1 + σ / √n2), where
● x1 = mean of sample 1
● x2 = mean of sample 2
● n1 = size of sample 1
● n2 = size of sample 2

16. What are hypothesis testing?
A statistical hypothesis is an assumption about a population parameter. This
assumption may or may not be true. Hypothesis testing refers to the formal
procedures used by statisticians to accept or reject statistical hypotheses.
Statistical Hypotheses
The best way to determine whether a statistical hypothesis is true would be to
examine the entire population. Since that is often impractical, researchers typically
examine a random sample from the population. If sample data are not consistent
with the statistical hypothesis, the hypothesis is rejected.
There are two types of statistical hypotheses.
Null hypothesis. The null hypothesis, denoted by Ho, is usually the hypothesis that
sample observations result purely from chance.
Alternative hypothesis. The alternative hypothesis, denoted by H1 or Ha, is the
hypothesis that sample observations are influenced by some non-random cause.
For example, suppose we wanted to determine whether a coin was fair and
balanced. A null hypothesis might be that half the flips would result in Heads and
half, in Tails. The alternative hypothesis might be that the number of Heads and Tails
would be very different. Symbolically, these hypotheses would be expressed as
● Ho: P = 0.5
● Ha: P ≠ 0.5
Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this
result, we would be inclined to reject the null hypothesis. We would conclude, based
on the evidence, that the coin was probably not fair and balanced.
Other terms related to Hypothesis Testing are:
Level of significance: Refers to the degree of significance in which we accept or
reject the null-hypothesis. 100% accuracy is not possible for accepting or rejecting a
hypothesis, so we therefore select a level of significance that is usually 5%.
Type I error: When we reject the null hypothesis, although that hypothesis was true.
Type I error is denoted by alpha. In hypothesis testing, the normal curve that shows
the critical region is called the alpha region.
Type II errors: When we accept the null hypothesis but it is false. Type II errors are
denoted by beta. In Hypothesis testing, the normal curve that shows the acceptance
region is called the beta region.
Power: Usually known as the probability of correctly accepting the null hypothesis.
1-beta is called power of the analysis.

One-tailed test: When the given statistical hypothesis is one value like H0: μ1 = μ2, it
is called the one-tailed test.
Two-tailed test: When the given statistics hypothesis assumes a less than or greater
than value, it is called the two-tailed test
17. How do you determine outliers?
The different ways of determining the outliers:
a. Simply sort your data sheet for each variable and then look for unusually high
or low values.
b. Graphing your data: Using Boxplots, Histogram and Scatterplots.
Boxplot: We use boxplots when we have groups in our data. Boxplots display asterisks
or other symbols on the graph to indicate explicitly when datasets contain
outliers.
Histogram: Histograms also emphasize the existence of outliers. Look for isolated bars in
the plot.
Scatterplot: We can use Scatterplot to determine outliers in a multivariate setting.
Isolated points from regression line gives us the outliers.
c. Using Z-Score: Z-scores can quantify the unusualness of an observation when
your data follow the normal distribution. Z-scores are the number of standard
deviations above and below the mean that each value falls. For example, a
Z-score of 2 indicates that an observation is two standard deviations above
the average while a Z-score of -2 signifies it is two standard deviations below the
mean. A Z-score of zero represents a value that equals the mean. Statistical rule
of thumb is if Z-Score for a specific data point is less than -4 or greater than 4 is
a suspected outlier.
d. Using IQR: You can use the interquartile range (IQR), several quartile values, and
an adjustment factor to calculate boundaries for what constitutes minor and
major outliers. Minor and major denote the unusualness of the outlier relative
to the overall distribution of values. Major outliers are more extreme. Analysts
also refer to these categorizations as mild and extreme outliers.
The IQR is the middle 50% of the dataset. It’s the range of values between the third
quartile and the first quartile (Q3 – Q1). We can take the IQR, Q1, and Q3 values
to calculate the following outlier fences for our dataset:
lower outer, lower inner, upper inner, and upper outer.
These fences determine whether data points are outliers and whether they are mild or
extreme.
Values that fall inside the two inner fences are not outliers.

e. Using Hypothesis Testing: You can use hypothesis tests to find outliers. Many
outlier tests exist, but I’ll focus on one to illustrate how they work. In this post, I
demonstrate Grubbs’ test, which tests the following hypotheses:
Null: All values in the sample were drawn from a single population that follows the same
normal distribution.
Alternative: One value in the sample was not drawn from the same normally distributed
population as the other values.
If the p-value for this test is less than your significance level, you can reject the null and
conclude that one of the values is an outlier.
f. Standard Deviation Method: We know that if the data follow normal
distribution then the data covers 99.7% of the points up to 3 standard deviation.
We can have our outliers calculated beyond that on both sides. Python Code for
this is as follows:

18. When do you reject or accept null hypothesis? List Steps.
Step 1: State the null hypothesis. When you state the null hypothesis, you also have to
state the alternate hypothesis. Sometimes it is easier to state the alternate hypothesis
first, because that’s the researcher’s thoughts about the experiment.
Step 2: Support or reject the null hypothesis. Several methods exist, depending on what
kind of sample data you have. For example, you can use the P-value method.
If you are able to reject the null hypothesis in Step 2, you can replace it with the
alternate hypothesis.
That’s it!
When to Reject the Null hypothesis?
Basically, you reject the null hypothesis when your test value falls into the rejection
region. There are four main ways you’ll compute test values and either support or reject
your null hypothesis. Which method you choose depends mainly on if you have a
proportion or a p-value.
(For details refer :
https://www.sagepub.com/sites/default/files/upmbinaries/40007_Chapter8.pdf)
19. What are Type I and Type II errors? How would you proceed to minimise them?
No hypothesis test is 100% certain. Because the test is based on probabilities, there
is always a chance of making an incorrect conclusion. When you do a hypothesis test,
two types of errors are possible: type I and type II. The risks of these two errors are
inversely related and determined by the level of significance and the power for the
test. Therefore, you should determine which error has more severe consequences
for your situation before you define their risks.

Type I error:
When the null hypothesis is true and you reject it, you make a type I error. The
probability of making a type I error is α, which is the level of significance you set for
your hypothesis test. An α of 0.05 indicates that you are willing to accept a 5%
chance that you are wrong when you reject the null hypothesis. To lower this risk,
you must use a lower value for α. However, using a lower value for alpha means
that you will be less likely to detect a true difference if one really exists.
Type II error:
When the null hypothesis is false and you fail to reject it, you make a type II error.
The probability of making a type II error is β, which depends on the power of the
test. You can decrease your risk of committing a type II error by ensuring your test
has enough power. You can do this by ensuring your sample size is large enough to
detect a practical difference when one truly exists.
The probability of rejecting the null hypothesis when it is false is equal to 1–β. This
value is the power of the test.
Avoiding Type-I Error:
Minimize the significance level (α) of Hypothesis Test: since we choose the significance
level. However, lowering the significance level may lead to a situation wherein the
results of the hypothesis test may not capture the true parameter or the true
difference of the test.
Avoiding Type-II Error:
a. Increase the sample size
b. Increase the significance level of Hypothesis Test.
20. What is Bias and Variance? How can we have optimum of both?
Bias is the difference between the average prediction of our model and the correct
value which we are trying to predict. Model with high bias pays very little attention
to the training data and oversimplifies the model. It always leads to high error on
training and test data.
Variance is the variability of model prediction for a given data point or a value which
tells us spread of our data. Model with high variance pays a lot of attention to
training data and does not generalize on the data which it hasn’t seen before. As a

result, such models perform very well on training data but has high error rates on
test data.
In supervised learning, underfitting happens when a model unable to capture the
underlying pattern of the data. These models usually have high bias and low
variance. It happens when we have very less amount of data to build an accurate
model or when we try to build a linear model with a nonlinear data. Also, these kind
of models are very simple to capture the complex patterns in data like Linear and
logistic regression.
In supervised learning, overfitting happens when our model captures the noise along
with the underlying pattern in data. It happens when we train our model a lot over
noisy dataset. These models have low bias and high variance. These models are very
complex like Decision trees which are prone to overfitting.
Bias Variance Tradeoff:
For any supervised algorithm, having a high bias error usually means it has low
variance error and vice versa. To be more specific, parametric or linear ML
algorithms often have a high bias but low variance. On the other hand,
non-parametric or non-linear algorithms have vice versa.
The goal of any ML model is to obtain a low variance and a low bias state, which is
often a task due to the parametrization of machine learning algorithms. Common
ways to achieve optimum Bias and Variance are:
a. By minimising total error
b. Using Bagging and resampling techniques
c. Adjusting minor values in Algorithms
(Refer :
https://hub.packtpub.com/heres-how-you-can-handle-the-bias-variance-trade-off-in-your-ml-
models/)

21. What is Confusion Matrix?
it is a performance measurement for machine learning classification problem where
output can be two or more classes. It is a table with 4 different combinations of
predicted and actual values.
Definition of the terms:
● Positive (P) : Observation is positive (for example: is an apple).
● Negative (N) : Observation is not positive (for example: is not an apple).
● True Positive (TP) : Observation is positive, and is predicted to be positive.
● False Negative (FN) : Observation is positive, but is predicted negative.
● True Negative (TN) : Observation is negative, and is predicted to be negative.
● False Positive (FP) : Observation is negative, but is predicted positive.
Classification Rate/Accuracy: Classification Rate or Accuracy is given by the relation:
However, there are problems with accuracy. It assumes equal costs for both kinds of
errors. A 99% accuracy can be excellent, good, mediocre, poor or terrible depending
upon the problem.
Recall/Sensitivity/True Positive Rate: Recall can be defined as the ratio of the total
number of correctly classified positive examples divide to the total number of
positive examples. High Recall indicates the class is correctly recognized (a small
number of FN).

Precision: To get the value of precision we divide the total number of correctly
classified positive examples by the total number of predicted positive examples. High
Precision indicates an example labelled as positive is indeed positive (a small number
of FP).
F-measure/F-stats/F1 Score: Since we have two measures (Precision and Recall) it
helps to have a measurement that represents both of them. We calculate an
F-measure which uses Harmonic Mean in place of Arithmetic Mean as it punishes the
extreme values more.
The F-Measure will always be nearer to the smaller value of Precision or Recall.
Specificity: Percentage of negative instances out of the total actual negative
instances. Therefore, denominator (TN + FP) here is the actual number of negative
instances present in the dataset. It is similar to recall but the shift is on the negative
instances. Like finding out how many healthy patients were not having cancer and
were told they don’t have cancer. Kind of a measure to see how separate the classes
are.
22. What are different ways of measuring the performance of different models?
The different ways are as follows:
● Confusion matrix
● Accuracy
● Precision
● Recall
● Specificity
● F1 score
● Precision-Recall or PR curve
● ROC (Receiver Operating Characteristics) curve
● PR vs ROC curve
Most of the ways we have already seen in the above question. We will discuss only
which are left.
PR Curve: It is the curve between precision and recall for various threshold values. In
the figure below we have 6 predictors showing their respective precision-recall curve

for various threshold values. The top right part of the graph is the ideal space where
we get high precision and recall. Based on our application we can choose the
predictor and the threshold value. PR AUC is just the area under the curve. The
higher its numerical value the better.
ROC Curve: ROC stands for receiver operating characteristic and the graph is plotted
against TPR and FPR for various threshold values. As TPR increases FPR also
increases. As you can see in the first figure, we have four categories and we want the
threshold value that leads us closer to the top left corner. Comparing different
predictors (here 3) on a given dataset also becomes easy as you can see in figure 2,
one can choose the threshold according to the application at hand. ROC AUC is just
the area under the curve, the higher its numerical value the better.
PR vs ROC Curve:
Both the metrics are widely used to judge a models performance.
Which one to use PR or ROC?
The answer lies in TRUE NEGATIVES.

Due to the absence of TN in the precision-recall equation, they are useful in
imbalanced classes. In the case of class imbalance when there is a majority of the
negative class. The metric doesn’t take much into consideration the high number of
TRUE NEGATIVES of the negative class which is in majority, giving better resistance
to the imbalance. This is important when the detection of the positive class is very
important.
Like to detect cancer patients, which has a high class imbalance because very few
have it out of all the diagnosed. We certainly don’t want to miss on a person having
cancer and going undetected (recall) and be sure the detected one is having it
(precision).
Due to the consideration of TN or the negative class in the ROC equation, it is useful
when both the classes are important to us. Like the detection of cats and dog. The
importance of true negatives makes sure that both the classes are given importance,
like the output of a CNN model in determining the image is of a cat or a dog.
23. What is correlation? What is the range that a correlation coefficient can have?
Correlation is a statistical technique that can show whether and how strongly pairs
of variables are related. For example, height and weight are related; taller people
tend to be heavier than shorter people.
The main result of a correlation is called the correlation coefficient (or "r"). It ranges
from -1.0 to +1.0. The closer r is to +1 or -1, the more closely the two variables are
related.
24. What is ROC Curve? What is AUC?
Refer to Question 16 for ROC Curve.
AUC: ROC curve is a performance measurement for classification problem at various
thresholds settings. ROC is a probability curve and AUC represents degree or
measure of separability. It tells how much model is capable of distinguishing
between classes. Higher the AUC, better the model is at predicting 0s as 0s and 1s as
1s. By analogy, Higher the AUC, better the model is at distinguishing between
patients with disease and no disease.
The ROC curve is plotted with TPR against the FPR where TPR is on y-axis and FPR is
on the x-axis.
25. What is Collinearity and Correlation?
Correlation measures the strength of linear relationship between two variables.
Collinearity: If in multiple regression analysis, one of the predictors is linearly
associated/dependent on other predictor, then this issue is known as collinearity.
For example, let’s consider the linear model
Y = αx1 + β1x1 + β2x2 ... (1)

If predictor x1 can be expressed as linear combination of x2, say, x1 = 3*x2
Then this is known as collinearity among the predictors. Note that there will be
perfect (or very high) correlation between the predictors as opposed to the
assumption of linear regression model (All predictors are assumed to be
independent).
Essentially it means that one of the independent variables is not really necessary to
the model because its effect/impact on the model is already captured by some of the
other variables. This variable is not contributing anything extra to the predictions
and can be removed. If we have true collinearity (perfect correlation as in the
example above), the one of the predictor is automatically deleted by some of the
software’s like R, other shows an error or warning for the same.
The effects of collinearity are seen in the variances of the parameter estimates, not
in the parameter estimates themselves.
Note: Collinearity implies correlation but not vice- versa.
26. What are the different types of Sampling?
Some of the Common sampling ways are as follows:
● Simple random sample: Every member and set of members has an equal
chance of being included in the sample. Technology, random number
generators, or some other sort of chance process is needed to get a simple
random sample.
Example—A teacher puts students' names in a hat and chooses without looking to
get a sample of students.
Why it's good: Random samples are usually fairly representative since they don't
favour certain members.
● Stratified random sample: The population is first split into groups. The
overall sample consists of some members from every group. The members
from each group are chosen randomly.
Example—A student council surveys 100100100 students by getting random samples
of 252525 freshmen, 252525 sophomores, 252525 juniors, and 252525
seniors.
Why it's good: A stratified sample guarantees that members from each group will be
represented in the sample, so this sampling method is good when we want
some members from every group.
● Cluster random sample: The population is first split into groups. The overall
sample consists of every member from some of the groups. The groups are
selected at random.

Example—An airline company wants to survey its customers one day, so they
randomly select 555 flights that day and survey every passenger on those
flights.
Why it's good: A cluster sample gets every member from some of the groups, so it's
good when each group reflects the population as a whole.
● Systematic random sample: Members of the population are put in some
order. A starting point is selected at random, and every nth member is
selected to be in the sample.
Example—A principal takes an alphabetized list of student names and picks a
random starting point. Every 20th
student is selected to take a survey.
27. What is confidence interval? What is its significance?
A confidence interval, in statistics, refers to the probability that a population
parameter will fall between two set values for a certain proportion of times.
Confidence intervals measure the degree of uncertainty or certainty in a sampling
method. A confidence interval can take any number of probabilities, with the most
common being a 95% or 99% confidence level.
28. What are the effects of the width of confidence interval?
● Confidence interval is used for decision making
● As the confidence level increases the width of the confidence interval also
increases
● As the width of the confidence interval increases, we tend to get useless
information also.
● Useless information – wide CI
● High risk – narrow CI
29. What is level of significance (Alpha)?
The significance level, also denoted as alpha or α, is a measure of the strength of
the evidence that must be present in your sample before you will reject the null
hypothesis and conclude that the effect is statistically significant. The researcher
determines the significance level before conducting the experiment.
The significance level is the probability of rejecting the null hypothesis when it is
true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a
difference exists when there is no actual difference. Lower significance levels
indicate that you require stronger evidence before you will reject the null
hypothesis.
Use significance levels during hypothesis testing to help you determine which
hypothesis the data support. Compare your p-value to your significance level. If the
p-value is less than your significance level, you can reject the null hypothesis and
conclude that the effect is statistically significant. In other words, the evidence in
your sample is strong enough to be able to reject the null hypothesis at the
population level.

30. What is Skewness and Kurtosis? What does it signify?
Skewness: It is the degree of distortion from the symmetrical bell curve or the
normal distribution. It measures the lack of symmetry in data distribution.
It differentiates extreme values in one versus the other tail. A symmetrical
distribution will have a skewness of 0.
There are two types of Skewness: Positive and Negative
Positive Skewness means when the tail on the right side of the distribution is longer
or fatter. The mean and median will be greater than the mode.
Negative Skewness is when the tail of the left side of the distribution is longer or
fatter than the tail on the right side. The mean and median will be less than the
mode.
So, when is the skewness too much?
The rule of thumb seems to be:
● If the skewness is between -0.5 and 0.5, the data are fairly symmetrical.
● If the skewness is between -1 and -0.5(negatively skewed) or between 0.5
and 1(positively skewed), the data are moderately skewed.
● If the skewness is less than -1(negatively skewed) or greater than 1(positively
skewed), the data are highly skewed.
Example
Let us take a very common example of house prices. Suppose we have house values
ranging from $100k to $1,000,000 with the average being $500,000.
If the peak of the distribution was left of the average value, portraying a positive
skewness in the distribution. It would mean that many houses were being sold for
less than the average value, i.e. $500k. This could be for many reasons, but we are
not going to interpret those reasons here.
If the peak of the distributed data was right of the average value, that would mean a
negative skew. This would mean that the houses were being sold for more than the
average value.

Kurtosis: Kurtosis is all about the tails of the distribution — not the peakedness or
flatness. It is used to describe the extreme values in one versus the other tail. It is
actually the measure of outliers present in the distribution.
High kurtosis in a data set is an indicator that data has heavy tails or outliers. If there
is a high kurtosis, then, we need to investigate why do we have so many outliers. It
indicates a lot of things, maybe wrong data entry or other things. Investigate!
Low kurtosis in a data set is an indicator that data has light tails or lack of outliers. If
we get low kurtosis(too good to be true), then also we need to investigate and trim
the dataset of unwanted results.
Mesokurtic: This distribution has kurtosis statistic similar to that of the normal
distribution. It means that the extreme values of the distribution are similar to that
of a normal distribution characteristic. This definition is used so that the standard
normal distribution has a kurtosis of three.
Leptokurtic (Kurtosis > 3): Distribution is longer, tails are fatter. Peak is higher and
sharper than Mesokurtic, which means that data are heavy-tailed or profusion of
outliers.
Outliers stretch the horizontal axis of the histogram graph, which makes the bulk of
the data appear in a narrow (“skinny”) vertical range, thereby giving the “skinniness”
of a leptokurtic distribution.
Platykurtic: (Kurtosis < 3): Distribution is shorter; tails are thinner than the normal
distribution. The peak is lower and broader than Mesokurtic, which means that data
are light-tailed or lack of outliers. The reason for this is because the extreme values
are less than that of the normal distribution.
31. What is Range and IQR? What does it signify?
Range: The range of a set of data is the difference between the highest and lowest
values in the set.
IQR(Inter Quartile Range): The interquartile range (IQR) is the difference between
the first quartile and third quartile. The formula for this is:
IQR = Q3 - Q1

The range gives us a measurement of how spread out the entirety of our data set is.
The interquartile range, which tells us how far apart the first and third quartile are,
indicates how spread out the middle 50% of our set of data is.
32. What is difference between Variance and Standard Deviation? What is its
significance?
The central tendency mean gives you the idea of average of the data points( i.e
centre location of the distribution)
And now you want to know how far are your data points from mean
So, here comes the concept of variance to calculate how far are your data points
from mean (in simple terms, it is to calculate the variation of your data points from
mean)
Standard deviation is simply the square root of variance . And standard deviation is
also used to calculate the variation of your data points
(And you may be asking, why do we use standard deviation , when we have variance.
Because, in order to maintain the calculations in same units i.e suppose mean is in
𝑐𝑚/ 𝑚, then variance is in 𝑐 𝑚2/ 𝑚2, whereas standard deviation is in 𝑐 𝑚/ 𝑚 , so we
use standard deviation most)
33. What is selection Bias? Types of Selection Bias?
Selection bias is the phenomenon of selecting individuals, groups or data for analysis
in such a way that proper randomization is not achieved, ultimately resulting in a
sample that is not representative of the population.
Understanding and identifying selection bias is important because it can significantly
skew results and provide false insights about a particular population group.
Types of selection bias include:
● Sampling bias: a biased sample caused by non-random sampling
● Time interval: selecting a specific time frame that supports the desired
conclusion. e.g. conducting a sales analysis near Christmas.
● Exposure: includes clinical susceptibility bias, protopathic bias, indication
bias. Read more here.

● Data: includes cherry-picking, suppressing evidence, and the fallacy of
incomplete evidence.
● Attrition: attrition bias is similar to survivorship bias, where only those that
‘survived’ a long process are included in an analysis, or failure bias, where
those that ‘failed’ are only included
● Observer selection: related to the Anthropic principle, which is a
philosophical consideration that any data we collect about the universe is
filtered by the fact that, in order for it to be observable, it must be
compatible with the conscious and sapient life that observes it.
Handling missing data can make selection bias worse because different methods
impact the data in different ways. For example, if you replace null values with the
mean of the data, you adding bias in the sense that you’re assuming that the data is
not as spread out as it might actually be.
34. What are the ways of handling missing Data?
● Delete rows with missing data
● Mean/Median/Mode imputation
● Assigning a unique value
● Predicting the missing values using Machine Learning Models
● Using an algorithm which supports missing values, like random forests
35. What are the different types of probability distribution? Explain with example?
The common Probability Distribution are as follows:
1. Bernoulli Distribution
2. Uniform Distribution
3. Binomial Distribution
4. Normal Distribution
5. Poisson Distribution
1. Bernoulli Distribution: A Bernoulli distribution has only two possible outcomes,
namely 1 (success) and 0 (failure), and a single trial. So the random variable X
which has a Bernoulli distribution can take value 1 with the probability of
success, say p, and the value 0 with the probability of failure, say q or 1-p.
Exmple: whether it’s going to rain tomorrow or not where rain denotes success and no
rain denotes failure and Winning (success) or losing (failure) the game.
2. Uniform Distribution: When you roll a fair die, the outcomes are 1 to 6. The
probabilities of getting these outcomes are equally likely and that is the basis of a
uniform distribution. Unlike Bernoulli Distribution, all the n number of possible
outcomes of a uniform distribution are equally likely.
Example: Rolling a fair dice.
3. Binomial Distribution: A distribution where only two outcomes are possible, such
as success or failure, gain or loss, win or lose and where the probability of
success and failure is same for all the trials is called a Binomial Distribution.
● Each trial is independent.

● There are only two possible outcomes in a trial- either a success or a
failure.
● A total number of n identical trials are conducted.
● The probability of success and failure is same for all trials. (Trials are
identical.)
Example: Tossing a coin.
4. Normal Distribution: Normal distribution represents the behaviour of most of the
situations in the universe (That is why it’s called a “normal” distribution. I
guess!). The large sum of (small) random variables often turns out to be normally
distributed, contributing to its widespread application. Any distribution is known
as Normal distribution if it has the following characteristics:
● The mean, median and mode of the distribution coincide.
● The curve of the distribution is bell-shaped and symmetrical about the
line x=μ.
● The total area under the curve is 1.
● Exactly half of the values are to the left of the center and the other half to
the right.
5. Poisson Distribution: A distribution is called Poisson distribution when the
following assumptions are valid:
● Any successful event should not influence the outcome of another
successful event.
● The probability of success over a short interval must equal the probability
of success over a longer interval.
● The probability of success in an interval approaches zero as the interval
becomes smaller.
Example: The number of emergency calls recorded at a hospital in a day.
36. What are statistical Tests? List Them.
Statistical tests are used in hypothesis testing. They can be used to:
● determine whether a predictor variable has a statistically significant
relationship with an outcome variable.
● estimate the difference between two or more groups.
Statistical tests assume a null hypothesis of no relationship or no difference between
groups. Then they determine whether the observed data fall outside of the range of
values predicted by the null hypothesis.
Common Tests in Statistics:
a. T-Test/Z-Test
b. ANOVA
c. Chi-Square Test
d. MANOVA

37. How do you calculate the sample size required?
You can use the margin of error (ME) formula to determine the desired sample size.
● t/z = t/z score used to calculate the confidence interval
● ME = the desired margin of error
● S = sample standard deviation
38. What are the different Biases associated when we sample?
Potential biases include the following:

● Sampling bias: a biased sample caused by non-random sampling
● Under coverage bias: sampling too few observations
● Survivorship bias: error of overlooking observations that did not make it past
a form of selection process.
39. How to convert normal distribution to standard normal distribution?
Standardized normal distribution has mean = 0 and standard deviation = 1
To convert normal distribution to standard normal distribution we can use the
formula:
X (standardized) = (x-µ) / σ
40. How to find the mean length of all fishes in a river?
● Define the confidence level (most common is 95%)
● Take a sample of fishes from the river (to get better results the number of fishes
> 30)
● Calculate the mean length and standard deviation of the lengths
● Calculate t-statistics
● Get the confidence interval in which the mean length of all the fishes should be.
41. What do you mean by degree of freedom?
● DF is defined as the number of options we have
● DF is used with t-distribution and not with Z-distribution
● For a series, DF = n-1 (where n is the number of observations in the series)
42. What do you think if DF is more than 30?
● As DF increases the t-distribution reaches closer to the normal distribution
● At low DF, we have fat tails
● If DF > 30, then t-distribution is as good as normal distribution
43. When to use t distribution and when to use z distribution?
● The following conditions must be satisfied to use Z-distribution
● Do we know the population standard deviation?
● Is the sample size > 30?
● CI = x (bar) – Z*σ/√n to x (bar) + Z*σ/√n
● Else we should use t-distribution
● CI = x (bar) – t*s/√n to x (bar) + t*s/√n
44. What is H0 and H1? What is H0 and H1 for two-tail test?
● H0 is known as null hypothesis. It is the normal case / default case.
For one tail test x <= µ
For two-tail test x = µ

● H1 is known as alternate hypothesis. It is the other case.
For one tail test x > µ
For two-tail test x <> µ
45. What is Degree of Freedom?
DF is defined as the number of options we have
DF is used with t-distribution and not with Z-distribution
For a series, DF = n-1 (where n is the number of observations in the series)
46. How to calculate p-Value?
Calculating p-value:
a. Using Excel:
I. Go to Data tab
II. Click on Data Analysis
III. Select Descriptive Statistics
IV. Choose the column
V. Select summary statistics and confidence level (0.95)
b. By Manual Method:
I. Find H0 and H1
II. Find n, x(bar) and s
III. Find DF for t-distribution
IV. Find the type of distribution – t or z distribution
V. Find t or z value (using the look-up table)
VI. Compute the p-value to critical value
47. What is ANOVA?
ANOVA expands to the analysis of variance, is described as a statistical technique
used to determine the difference in the means of two or more populations, by
examining the amount of variation within the samples corresponding to the amount
of variation between the samples. It bifurcates the total amount of variation in the
dataset into two parts, i.e. the amount ascribed to chance and the amount ascribed
to specific causes.
It is a method of analysing the factors which are hypothesised or affect the
dependent variable. It can also be used to study the variations amongst different
categories, within the factors, that consist of numerous possible values. It is of two
types:
One way ANOVA: When one factor is used to investigate the difference amongst
different categories, having many possible values.
Two way ANOVA: When two factors are investigated simultaneously to measure the
interaction of the two factors influencing the values of a variable.
48. What is ANCOVA?

ANCOVA stands for Analysis of Covariance, is an extended form of ANOVA, that
eliminates the effect of one or more interval-scaled extraneous variable, from the
dependent variable before carrying out research. It is the midpoint between ANOVA
and regression analysis, wherein one variable in two or more population can be
compared while considering the variability of other variables.
When in a set of independent variable consist of both factor (categorical
independent variable) and covariate (metric independent variable), the technique
used is known as ANCOVA. The difference in dependent variables because of the
covariate is taken off by an adjustment of the dependent variable’s mean value
within each treatment condition.
This technique is appropriate when the metric independent variable is linearly
associated with the dependent variable and not to the other factors. It is based on
certain assumptions which are:
There is some relationship between dependent and uncontrolled variable.
The relationship is linear and is identical from one group to another.
Various treatment groups are picked up at random from the population.
Groups are homogeneous in variability.
49. What are difference between ANOVA and ANCOVA?
The points given below are substantial so far as the difference between AOVA and
ANCOVA is concerned:
● The technique of identifying the variance among the means of multiple groups
for homogeneity is known as Analysis of Variance or ANOVA. A statistical
process which is used to take off the impact of one or more metric-scaled
undesirable variable from dependent variable before undertaking research is
known as ANCOVA.
● While ANOVA uses both linear and non-linear model. On the contrary, ANCOVA
uses only linear model.
● ANOVA entails only categorical independent variable, i.e. factor. As against this,
ANCOVA encompasses a categorical and a metric independent variable.
● A covariate is not taken into account, in ANOVA, but considered in ANCOVA.
● ANOVA characterises between group variations, exclusively to treatment. In
contrast, ANCOVA divides between group variations to treatment and covariate.
● ANOVA exhibits within group variations, particularly individual differences.
Unlike ANCOVA, that bifurcates within group variance in individual differences
and covariate.
50. What are t and z scores? Give Details.
T-Score vs. Z-Score: Overview
A z-score and a t score are both used in hypothesis testing.
T-score vs. z-score: When to use a t score
The general rule of thumb for when to use a t score is when your sample:

Has a sample size below 30,
Has an unknown population standard deviation.
You must know the standard deviation of the population and your sample size
should be above 30 in order for you to be able to use the z-score. Otherwise, use the
t-score.
Z-score
Technically, z-scores are a conversion of individual scores into a standard form. The
conversion allows you to more easily compare different data. A z-score tells you how
many standard deviations from the mean your result is. You can use your knowledge
of normal distributions (like the 68 95 and 99.7 rule) or the z-table to determine
what percentage of the population will fall below or above your result.
The z-score is calculated using the formula:
● z = (X-μ)/σ
Where:
● σ is the population standard deviation and
● μ is the population mean.
● The z-score formula doesn’t say anything about sample size; The rule of thumb
applies that your sample size should be above 30 to use it.
T-score
Like z-scores, t-scores are also a conversion of individual scores into a standard form.
However, t-scores are used when you don’t know the population standard deviation;
You make an estimate by using your sample.
● T = (X – μ) / [ s/√(n) ]
Where:
● s is the standard deviation of the sample.
If you have a larger sample (over 30), the t-distribution and z-distribution look pretty
much the same.
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