Step 1: Find the mean of the data.
The given observations are: $x_i=3,6,2,1,7,5$. Total number of observations, $n=6$.
Mean $\overline{x}=\frac{\sum x_i}{n}=\frac{3+6+2+1+7+5}{6}=\frac{24}{6}=4$.

Step 2: Find the deviations from the mean and their squares.

Step 3: Calculate the standard deviation ($\sigma$).
$$\sigma =\sqrt{\frac{\sum {(x_i-\overline{x})}^2}{n}}=\sqrt{\frac{28}{6}}=\sqrt{4.667}\approx 2.16$$

Step 1: Find the mean of the data.
The scores are: $40,38,42,60,72,54$. Number of observations, $n=6$.
Mean $\overline{x}=\frac{40+38+42+60+72+54}{6}=\frac{306}{6}=51$.

Step 2: Find the deviations from the mean and their squares.

Step 3: Calculate the standard deviation ($\sigma$).
$$\sigma =\sqrt{\frac{\sum {(x_i-\overline{x})}^2}{n}}=\sqrt{\frac{902}{6}}=\sqrt{150.33}\approx 12.26$$

Step 1: Understand the property of Mean and Standard Deviation.
Let the original set of numbers be $x_i$. The new set of numbers is generated using a linear transformation $y_i=a{x}_i+b$. Here, $a=2$ and $b=5$. So, $y_i=2x_i+5$.

Step 2: Relationship for the Mean.
The mean changes by the same exact scale and origin shifts applied to the individual data points.
$$\overline{y}=a\overline{x}+b=2\overline{x}+5$$ Therefore, the New Mean is obtained by multiplying the old mean by 2 and adding 5. (From Q1, old mean was 4. New mean will be $2\left(4\right)+5=13$).

Step 3: Relationship for the Standard Deviation.
Standard Deviation is a measure of spread. It is independent of the change of origin (adding a constant $b$ does not affect the spread), but it is affected by a change of scale (multiplying by a constant $a$).
$${\sigma }_y=\left|a\right|{\sigma }_x=\left|2\right|{\sigma }_x=2{\sigma }_x$$

Percentile and Percentile Rank are related concepts but describe different aspects of relative standing in a dataset.

• Percentile: A percentile is a value (or score) below which a certain percentage of the observations fall. For example, if a score of 75 represents the 90th percentile, it means that 90% of the scores in the entire distribution are strictly less than 75. It points to the actual data value.
• Percentile Rank: The percentile rank of a specific score is the percentage of scores in its frequency distribution that are equal to or lower than it. For instance, if you score an 85 on a test and your percentile rank is 80, it means you scored better than or equal to 80% of the students who took the test. It points to a percentage position.

Measures of dispersion measure the spread of values in a dataset. Quartile deviation, Mean deviation, and Standard deviation are all universally recognized measures of dispersion.

Step 1: Understand the property of variance.
If a constant $k$ is multiplied to each observation of a dataset, the new variance is multiplied by $k^2$.
$$\text{New Variance}=k^2\times \text{Original Variance}$$

Step 2: Calculate the new variance.
Here, original variance = $18$ and $k=3$.
$$\text{New Variance}=3^2\times 18=9\times 18=162$$

Step 1: Find the median.
The data arranged in ascending order is: $4,8,11,16,18$. There are $n=5$ (odd) observations. The median is the middle value, which is the 3rd observation. Median $M=11$.

Step 2: Find the absolute deviations from the median $|x_i-M|$.

• $|4-11|=7$
• $|8-11|=3$
• $|11-11|=0$
• $|16-11|=5$
• $|18-11|=7$

Step 3: Calculate the Mean Deviation.
$$M.D.\left(\mathrm{Median}\right)=\frac{\sum |x_i-M|}{n}=\frac{7+3+0+5+7}{5}=\frac{22}{5}=4.4$$

If each value in a dataset is multiplied by a constant factor $k$, the spread of the data is scaled by the absolute value of that factor. Therefore, the new standard deviation becomes $\left|k\right|$ times the original standard deviation.

Step 1: Arrange the data in ascending order.
The sorted scores are: $14,18,20,21,22,25,26,28,30,32$.
Total number of observations, $N=10$.

Step 2: Find the rank ($R$) of the score 25.
By looking at the ordered list, the score $25$ is at the 6th position. Thus, $R=6$.

Step 3: Calculate the Percentile Rank (PR).
The formula for Percentile Rank in an individual series is: $$PR=\frac{100}{N}(R-0.5)$$ Substitute $R=6$ and $N=10$: $$PR=\frac{100}{10}(6-0.5)=10\times 5.5=55$$ So, the score lies at the 55th percentile.

By definition, standard deviation is the positive square root of variance. Therefore, variance ($v$) is the square of standard deviation ($\sigma$).

One of the fundamental mathematical properties of the median is that the sum of the absolute deviations of the observations from the median is minimal. Consequently, the mean deviation is the smallest when it is calculated about the median.

Step 1: Understand the property of standard deviation.
If each observation is divided by a constant $k$, the new standard deviation is obtained by dividing the original standard deviation by the absolute value of $k$. Standard deviation is always a non-negative quantity. $$\text{New S.D.}=\frac{\text{Original S.D.}}{\left|k\right|}$$

Step 2: Calculate the new S.D.
Here, original S.D. = $12$ and $k=-3$.
$$\text{New S.D.}=\frac{12}{|-3|}=\frac{12}{3}=4$$

Range is the simplest measure of dispersion. It represents the total spread of a dataset and is defined purely as the difference between the maximum (highest) and minimum (lowest) values in the dataset.

By the algebraic property of the arithmetic mean, the algebraic sum of deviations of a set of observations from their mean is always identically zero.
Mathematically: $$\sum _{i=1}^n(x_i-\overline{x})=\sum _{i=1}^n{x}_i-\sum _{i=1}^n\overline{x}=n\overline{x}-n\overline{x}=0$$

The first $n$ natural numbers are $1,2,3,...,n$.

Step 1: Calculate the Mean.
The sum of the first $n$ natural numbers is given by: $$\sum x=\frac{n(n+1)}{2}$$ Mean ($\overline{x}$) = $\frac{\sum x}{n}=\frac{n(n+1)}{2n}=\frac{n+1}{2}$.

Step 2: Calculate the Variance (${\sigma }^2$).
The sum of the squares of the first $n$ natural numbers is given by: $$\sum x^2=\frac{n(n+1)(2n+1)}{6}$$ Variance ${\sigma }^2=\frac{\sum x^2}{n}-{\left(\overline{x}\right)}^2$
$${\sigma }^2=\frac{n(n+1)(2n+1)}{6n}-{\left(\frac{n+1}{2}\right)}^2$$ $${\sigma }^2=\frac{(n+1)(2n+1)}{6}-\frac{{(n+1)}^2}{4}$$ Factor out $\frac{n+1}{2}$: Find a common denominator for the terms in the bracket (which is 6):

Step 3: Calculate the Standard Deviation ($\sigma$).
Standard deviation is the square root of the variance. $$\sigma =\sqrt{\frac{n^2-1}{12}}$$

Step 1: Arrange the marks in ascending order.
Sorted list: $30,35,47,49,50,55,59,60,62,65,65,65,67,68,70,72,78,80,88,95$.
Total number of observations, $N=20$.

Step 2: Determine the rank of the score 65.
Looking at the ordered list, the score $65$ appears multiple times. It occupies positions 10th, 11th, and 12th. When a value repeats, we take the average of these ranks.
Rank $R=\frac{10+11+12}{3}=11$.

Step 3: Calculate the Percentile Rank (PR).
Formula: $$PR=\frac{100}{N}(R-0.5)$$ $$PR=\frac{100}{20}(11-0.5)=5\times 10.5=52.5$$

Step 1: Calculate the Cumulative Frequency (CF) for the distribution.

Total number of students, $N=35$.

Step 2: Determine parameters for the score 55.
For a discrete frequency distribution, the effective rank position ($R$) representing the midpoint of the data bundle corresponding to the required score is given by:
$$R\; =\; c\backslash mathit\{f\}\_\{\backslash text\{below\}\}\; +\; 0.5\; \backslash times\; f$$ Where:

• $c{f}_{\text{below}}$ is the cumulative frequency of the scores strictly below 55. Here, $c{f}_{\text{below}}=23$ (the $cf$ up to score 50).
• $f$ is the frequency of the score 55 itself. Here, $f=6$.

Step 3: Calculate the Percentile Rank.
$$PR=\frac{100}{N}\times R=\frac{100}{35}\times 26=\frac{2600}{35}\approx 74.285$$

Step 1: Calculate the Cumulative Frequency (CF) table.

Total number of students, $N=32$.

Step 2: Calculate Percentile Rank (PR) using formula $PR=\frac{100}{N}(c{f}_{\text{below}}+0.5f)$

For Score 20:
$c{f}_{\text{below}}=14$, $f=6$.
$$PR=\frac{100}{32}(14+0.5\times 6)=\frac{100}{32}(14+3)=\frac{1700}{32}=53.125$$

For Score 22:
$c{f}_{\text{below}}=20$, $f=7$.
$$PR=\frac{100}{32}(20+0.5\times 7)=\frac{100}{32}(20+3.5)=\frac{2350}{32}=73.4375$$

For Score 25:
$c{f}_{\text{below}}=27$, $f=3$.
$$PR=\frac{100}{32}(27+0.5\times 3)=\frac{100}{32}(27+1.5)=\frac{2850}{32}=89.0625$$

Step 1: Arrange the marks in ascending order.
Sorted Data: $12,15,20,28,30,40,50$.
Number of observations, $n=7$.

Step 2: Calculate the First Quartile ($Q_1$) and Third Quartile ($Q_3$).
$$Q_1={\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}={\left(\frac{7+1}{4}\right)}^{\text{th}}\text{term}=2^{\text{nd}}\text{term}=15$$ $$Q_3=3{\left(\frac{n+1}{4}\right)}^{\text{th}}\text{term}=3\left(2\right)\text{th term}=6^{\text{th}}\text{term}=40$$

Step 3: Calculate the Quartile Deviation (Q.D.) and its Coefficient.
$$Q.D.=\frac{Q_3-Q_1}{2}=\frac{40-15}{2}=\frac{25}{2}=12.5$$ $$\text{Coefficient of Q.D.}=\frac{Q_3-Q_1}{Q_3+Q_1}=\frac{40-15}{40+15}=\frac{25}{55}=\frac{5}{11}\approx 0.4545$$

Step 1: Identify that frequencies are cumulative.
The given "Number of Students" row is strictly increasing and represents a "less than" cumulative distribution. We must convert it to a standard frequency distribution by finding the individual class frequencies ($f_i$).

• $0-10:12$
• $10-20:30-12=18$
• $20-30:65-30=35$
• $30-40:107-65=42$
• $40-50:157-107=50$
• $50-60:202-157=45$
• $60-70:222-202=20$
• $70-80:230-222=8$

Total frequency $N=230$.

Step 2: Prepare calculation table using the Step Deviation Method.
Let assumed mean $A=45$ and class width $h=10$. We calculate $y_i=\frac{x_i-A}{h}$.

Step 3: Calculate Variance (${\sigma }^2$) and S.D. ($\sigma$).

Step 1: Write down the given incorrect data parameters.
Number of candidates, $n=200$
Incorrect Mean ${\overline{x}}_{\text{wrong}}=40$
Incorrect S.D. ${\sigma }_{\text{wrong}}=15$
Wrong observation = 34, Correct observation = 43.

Step 2: Find the correct mean.
Incorrect sum of observations: $\sum x_{\text{wrong}}=n\times {\overline{x}}_{\text{wrong}}=200\times 40=8000$.
Correct sum: $\sum x_{\text{correct}}=8000-34+43=8009$.
Correct Mean: ${\overline{x}}_{\text{correct}}=\frac{8009}{200}=40.045$.

Step 3: Find the correct sum of squares.
We use the variance formula: ${\sigma }^2=\frac{\sum x^2}{n}-{\left(\overline{x}\right)}^2$
$${15}^2=\frac{\sum x_{\text{wrong}}^2}{200}-{40}^2$$ $$225\; =\; \backslash frac\{\backslash sum\; x\_\{\backslash text\{wrong\}\}^2\}\{200\}\; -\; 1600\; \backslash implies\; \backslash frac\{\backslash sum\; x\_\{\backslash text\{wrong\}\}^2\}\{200\}\; =\; 1825\; \backslash implies\; \backslash sum\; x\_\{\backslash text\{wrong\}\}^2\; =\; 1825\; \backslash times\; 200\; =\; 365000$$ $$\sum x_{\text{correct}}^2=365000-{34}^2+{43}^2=365000-1156+1849=365693$$

Step 4: Calculate the correct S.D.
$$\text{Correct Variance}\left({\sigma }_{\text{correct}}^2\right)=\frac{\sum x_{\text{correct}}^2}{n}-{\left({\overline{x}}_{\text{correct}}\right)}^2$$ $${\sigma }_{\text{correct}}^2=\frac{365693}{200}-{\left(40.045\right)}^2=1828.465-1603.602=224.863$$ $${\sigma }_{\text{correct}}=\sqrt{224.863}\approx 14.995$$

Initial incorrect values:
$n=20$, $\overline{x}=10$, $\sigma =2$.
Incorrect Sum: $\sum x=20\times 10=200$.
Incorrect Sum of Squares: ${\sigma }^2=\frac{\sum x^2}{20}-{10}^2\Rightarrow 2^2=\frac{\sum x^2}{20}-100\Rightarrow \sum x^2=104\times 20=2080$.

Case (i): The wrong observation (8) is omitted.
New number of observations, $n^{\text{'}}=20-1=19$.
New Sum: $\sum x^{\text{'}}=200-8=192$.
New Mean: ${\overline{x}}^{\text{'}}=\frac{192}{19}\approx 10.105$.
New Sum of Squares: $\sum {\left(x^{\text{'}}\right)}^2=2080-8^2=2080-64=2016$.
New Variance: $${\left({\sigma }^{\text{'}}\right)}^2=\frac{2016}{19}-{\left(\frac{192}{19}\right)}^2=\frac{2016\times 19-36864}{361}=\frac{38304-36864}{361}=\frac{1440}{361}\approx 3.9889$$ New S.D.: ${\sigma }^{\text{'}}=\sqrt{3.9889}\approx 1.997$.

Case (ii): The wrong observation (8) is replaced by 12.
Number of observations remains, $n^{"}=20$.
New Sum: $\sum x^{"}=200-8+12=204$.
New Mean: ${\overline{x}}^{"}=\frac{204}{20}=10.2$.
New Sum of Squares: $\sum {\left(x^{"}\right)}^2=2080-8^2+{12}^2=2080-64+144=2160$.
New Variance: $${\left({\sigma }^{"}\right)}^2=\frac{2160}{20}-{\left(10.2\right)}^2=108-104.04=3.96$$ New S.D.: ${\sigma }^{"}=\sqrt{3.96}\approx 1.9899$.