Descriptive statistics:
Hence, When the nature of the population from which samples is drawn is not known to be normally distributed the data can be analyzed with the help of Nonparametric statistics.
Central tendency: Central tendency provides descriptive representation of entire data set which reflects the entire data set. Mean, Mode and median are measures of central tendency.
Mean: Mean (average) is the sum of all quantities divided by no of quantities of the data set.
Mode: It is the most repeated value in the data set.
Median: Median is the middlemost value of the data set.
Dispersion: This tells about the scatterings of data around the mean. It gives information about how stretched or squeezed the distribution of data are. Range, std deviation, percentile, etc. are used to measure dispersion.
Shape of a probability distribution:
Skewness:
The skewness is a measure of the asymmetry of the probability distribution assuming a unimodal distribution (assuming there is only one peak in the distribution of values) .
We can say that the skewness indicates how much our underlying distribution deviates from the normal distribution since the normal distribution has skewness 0. Generally, we have three types of skewness.
Important Points
Positive skew: When the right tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the left tail. In this case, we can use also the term “rightskewed” or “righttailed”. and the median is less than the mean.
If a test shows, a rightskewed distribution, this means most of the students are low scorers who scored below average and there is a very less number of students who scored very high marks. We can conclude that the test was very difficult and there are very few good students who scored high marks and these students can be considered outliers as they are not following the general trend.
Kurtosis
â€‹
Criticism is the practice of judging the merits and faults of something. Criticism is often presented as something unpleasant, but there are friendly criticisms, amicably discussed, and some people find great pleasure in criticism.
1) External criticism:
2) Internal criticism:
3) MetaAnalysis:
4) Trend Analysis:
Conclusion: From the above discussion, it is clear that MetaAnalysis is the review process that uses statistical methods to synthesize the results of independently conducted prior studies.
A researcher given two tests one in English language and another in Mathematics. He/She finds the mean and standard deviation of each of the two groups n = 100, as follows:

English 
Mathematics 
Mean 
50 
60 
Standard deviation 
10 
12 
Assuming the curve for the two distributions overlapping, how many students in English language are below the mean of those in Mathematics?
The curve for the two distributions is
We know that One standard deviation, or one sigma, plotted above or below the average value on that normal distribution curve, would define a region that includes 68 percent of all the data points. Two sigmas above or below would include about 95 percent of the data, and three sigmas would include 99.7 percent. This can be understood by the graph given below:
Therefore, the value between the means of English and Mathematics is µ + 1\(\sigma\) = 34.1 %
The number of students in the shaded region given in the above graph = \({Percentage\ of\ the\ shaded\ region \over 100} \times total\ number\ of\ students\ (n)\)
⇒ the number of students in the shaded region = \({34.1 \over100} \times 100\)
⇒ the number of students in the shaded region = 34.1 ≈ 34 (No. of students is discrete value)
Important Points
We need to find the number of students in the English language below the mean of those in Mathematics i.e the portion below the mean of mathematics.
⇒ 50% + 34 = 50 + 34 = 84
Hence 84 students in English language are below the mean of those in Mathematics.
Measures of central tendency provides a single value that indicates the general magnitude of the data and this single value provides information about the characteristics of the data by identifying the value at or near the central location of the data.
Key Points The three measures of central tendency that we will be discussing are: Mean or Arithmetic mean, Median and Mode.
Properties of Mean:
Thus, it is concluded that Arithmetic mean is most affected by extreme scores.
Hint
Normal Distribution is an arrangement of values in which most of the values are in the middle and the rest are symmetrically distributed at the end. This observed in the normal probability curve. Normal Probability Curve is a bellshaped curve having the highest point at the mean which is symmetrical along the vertical line drawn at the mean. It is used to interpret the percentile or the percentage of no of cases for the respective value of z scores.
Terms used to interpret normal probability curve are:
Mean: Mean (average) is the sum of all quantities divided by no of quantities.
Standard deviation: It is used to measure how deviated or dispersed the numbers are with respect to the mean in the same set of data.
Z score: Z score is a value calculated through the above formula, used to interpret percentile of a given score or value.
Percentile Rank: It is measured to find how many values are below the score. It basically rank of the score.
For e.g. If a student is having 85 percentile rank that means 85% of students are below him.
In question, it is given the percentile rank of the student on a test was found out to be 67. This means that the student had a test score greater than or equal to 67% of the reference population or we can say the student scored 67% marks. Conversely, 33% of students scored equal to or higher than the individual tested.
Ans. Option 4
Mean: Mean (average) is sum of all quantities divided by no of quantities.
Standard deviation: It is used to measure how deviated or dispersed the numbers are with respect to the mean in the same set of data.
Variance: Variance is a measure of how far a set of data (numbers) are spread out from their mean (average) value. Variance is square of standard deviation.
Reliability: is the state of being consistent or reliable. It is the overall consistency of the test. A test is reliable if different trials give same results or different parts of test give same results.
Reliability coefficient is a numerical term used to show how reliable the test is. The Cronbach alpha coefficients (α )were calculated to test the test’s reliability.
Reliability Coefficient (α ) = (N / (N1)) ( (Total Variance  sum of Individual Variance) / Total Variance)
⇒ \(Î± = {N \over (N1)}[{(Vi {Vt)} \over Vt}]\)
Where, N is number of Test items,
Total variance (V_{i}) = N * variance of test, sum of individual variance(V_{t})=N2*M(1M)
Given: N = 50, M = 0.48, Vi= 50*81, Vt= N^{2}*M(1M) = 50*50*0.48(10.48)=50*24*0.52 = 624
To find: Reliability coefficient (α )
Calculation:
\(\)\(Î± = {N \over (N1)}[{(Vi {Vt)} \over Vt}]\)
\( Î± = {50 \over 49}[{4050 {624} \over 4050}] \)
\(Î± = {50 \over 49}[{ 3426\over 4050}] \)
α = 0.86
Hence, A multiplechoice test consists of 50 test items. The difficulty values of items vary in a narrow range with an average value of 0.48. The test gave a variance of 81 when administered to a â€‹group of students. The reliability coefficient of the test would be 0.86.
Additional Information
Types of reliability tests:
Testretest reliability:
Parallel forms reliability:
Internal consistency:
Ways to measures :
In statistics,we have various types of scores like raw score, t score and z score.
A raw score is an original data which has not been transformed or altered. For e.g. the original marks obtained by a student in a test as opposed to that score after changed to a standard score or percentile rank.
Tscore is a conversion of raw score to the standard score which is based on the mean and standard deviation of mean.
For small sample(n<30), t score is calculated and for larger value z score is called.
Z score: Z score is a value calculated through the formula Z = \((X  \bar{X}) \over Ïƒ\), used to interpret percentile of a given score or value from normal probability curve.
Given: mean deviation\((X  \bar{X}) \)â€‹â€‹= 10, std deviation = 5.
To find: T score
Formula: Tscore = 50 + (zscore) x10
Z = \((X  \bar{X}) \over Ïƒ\)
Calculation:
Z = \((X  \bar{X}) \over Ïƒ\)
= 10/5
= 2
Tscore = 50 + (zscore) x10
= 50 + 2*10
= 70
Ans. Option 3
Mean: Mean (average) is sum of all quantities divided by no of quantities.
Standard deviation: It is used to measure how deviated or dispersed the numbers are with respect to the mean in the same set of data.
Confidence level: Confidence level means the percentage of the population that we are sure the estimate will fall between the specified range. It is a meaasue of reliability of test
They are used in estimating the accuracy level of any data.
e.g. Confidence level of 95% means 95% of population, we are sure that the estimated value will fall between the upper and lower values.
\(x =(\bar{X}) \pm Z{Ïƒ \over \sqrt{N}}\)
\((\bar{X})\) is sample mean
σ is the standard deviation
z is the value for the confidence level
N is sample size
Given: \((\bar{X})\)=100, σ = 20, N=100, since it is given at 0.01% is level of confidence i.e. 100(10.01) = 99% confidence level we can say the z value is 2.58 from below table.
z value for the confidence level  
80%  1.28 
90%  1.645 
95%  1.96 
98%  2.33 
99%  2.58 

To find range of population mean
Formula: Range for population mean \(x =(\bar{X}) \pm Z{Ïƒ \over \sqrt{N}}\)
Calculation:
\(x =(\bar{X}) \pm Z{Ïƒ \over \sqrt{N}}\)
= \((100) \pm 2.58{20 \over \sqrt{100}}\)
= 100 \( \pm\) 5.16
= 1005.16 or 100+5.16
= 94.84 or 105.16
= 94.8 or 105.2 appprox

Descriptive statistics:
Hence, When the nature of the population from which samples is drawn is not known to be normally distributed the data can be analyzed with the help of Nonparametric statistics.
Central tendency: Central tendency provides descriptive representation of entire data set which reflects the entire data set. Mean, Mode and median are measures of central tendency.
Mean: Mean (average) is the sum of all quantities divided by no of quantities of the data set.
Mode: It is the most repeated value in the data set.
Median: Median is the middlemost value of the data set.
Dispersion: This tells about the scatterings of data around the mean. It gives information about how stretched or squeezed the distribution of data are. Range, std deviation, percentile, etc. are used to measure dispersion.
Shape of a probability distribution:
Skewness:
The skewness is a measure of the asymmetry of the probability distribution assuming a unimodal distribution (assuming there is only one peak in the distribution of values) .
We can say that the skewness indicates how much our underlying distribution deviates from the normal distribution since the normal distribution has skewness 0. Generally, we have three types of skewness.
Important Points
Positive skew: When the right tail of the histogram of the distribution is longer and the majority of the observations are concentrated on the left tail. In this case, we can use also the term “rightskewed” or “righttailed”. and the median is less than the mean.
If a test shows, a rightskewed distribution, this means most of the students are low scorers who scored below average and there is a very less number of students who scored very high marks. We can conclude that the test was very difficult and there are very few good students who scored high marks and these students can be considered outliers as they are not following the general trend.
Kurtosis
â€‹
Normal Distribution is an arrangement of values in which most of the values are in the middle and rest are symmetrically distributed at the end. This observed in normal probability curve. Normal Probability Curve is a bell shaped curve having highest point at mean which is symmetrical along the vertical line drawn at the mean. It is used to interpret the percentile or the percentage of no of cases for respective value of z scores.
Terms used to interpret normal probability curve are:
Mean: Mean (average) is sum of all quantities divided by no of quantities.
Standard deviation: It is used to measure how deviated or dispersed the numbers are with respect to the mean in the same set of data.
Percentile Rank: It is measure to find how many values are below the score. It basically rank of the score.
Z score: Z score is a value calculated through the above formula, used to interpret percentile of a given score or value.
Important Points
To calculate: the percentile rank
Given: score(x) = 48, mean (xÌ„) = 60 and standard deviation = 12
Formula: Z = \((X  \bar{X}) \over Ïƒ\)
Calculation:
Step 1:
Calculate z scores
z = (4860)/12 = 1
Step 2:
Calculate the percentile rank:
From Normal Probability Curve, the total percentage of cases:
So total percentage of cases till the score 48 = 0.14% + 2.14% + 13.59% ( from the NPC given above)
So total percentage of cases till the score 48 = 15.87%
Therefore no of cases falling above the score 48 = 100% 15.87% = 84.13% of N (N=300)
⇒ no of cases falling above the score 48 = 84.13% of 300 = 252
Answer: No of cases falling above the score 48 = 252
Normal Distribution is an arrangement of values in which most of the values are in the middle and rest are symmetrically distributed at the end. This observed in normal probability curve. Normal Probability Curve is a bell shaped curve having highest point at mean which is symmetrical along the vertical line drawn at the mean. It is used to interpret the percentile or the percentage of no of cases for respective value of z scores.
Terms used to interpret normal probability curve are:
Mean: Mean (average) is sum of all quantities divided by no of quantities.
Standard deviation: It is used to measure how deviated or dispersed the numbers are with respect to the mean in the same set of data.
Percentile Rank: It is measure to find how many values are below the score. It basically rank of the score.
Z score: Z score is a value calculated through the above formula, used to interpret percentile of a given score or value.
Important Points
To calculate: the percentile rank
Given: z score = 1z
Calculation:
Calculate the percentile rank:
From Normal Probability Curve, the total percentage of cases:
from 0 to 1 σ = 0.14% + 2.14% + 13.59% ( from the NPC diagram given below)
So total percentage of cases = 15.87% = 16% approx.
Therefore, the percentile rank of students scoring at −1Z is 16
Significance of mean
Standard error of mean: The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation. In statistics, a sample mean deviates from the actual mean of a population—this deviation is the standard error of the mean.
In the question, we have given:
Step 1: Calculate standard error of mean
⇒ \(\sigma_{E} = \frac{\sigma}{\sqrt{N}}\)
⇒ \(\sigma_{E} = \frac{8}{\sqrt{256}}\) = 8/16 = 1/2
Step 2: locate zvalue of 0.95% ( âˆµ the probability is 0.95) from the z table i.e 1.96
Step 3: Calculate confidence interval at 0.95
= X + z × \(\sigma_{E}\) , X  z × \(\sigma_{E}\)
= 42 + 1.96 × 1/2, 42 + 1.96 × 1/2
= 42 + 0.98, 42  0.98
= 42.98, 41.02
upper limit = 42.98
The percentile rank
Let's understand the concept with the example given below:
Hence, the shape of the curve showing the distribution of percentile ranks is rectangular.
Additional Information
Normal distribution:
Leptokurtic:
Bimodal distribution:
Key Points
Normal Probability Curve: It is a bellshaped curve having highest point at mean which is symmetrical along the vertical line drawn at the mean. It is used to interpret the percentile or the percentage of no of cases for respective value of z scores.
Mean: Mean (average) is sum of all quantities divided by no of quantities.
Standard deviation: It is used to measure how deviated or dispersed the numbers are with respect to the mean in the same set of data.
Percentile Rank: It is a measure to find how many values are below the score. It is basically the rank of the score.
Z score: Z score is a value calculated through the above formula, used to interpret percentile of a given score or value.
Important Points
Given: n = 360, Mean Score (xÌ„) = 45 , σ = 9
To find: percentage of students who scored outside the limits of scores 36 – 54
Formula: Z = (x−xÌ„)/σ
Calculation:
To find out the percentage of students in a limit, we need to calculate z score
At x=36, Z = (36−45)/9 = 1
At x= 54, Z = (54−45)/9 = 1