Step 1: Establishing Variables and Theoretical Foundation

Let the eight observations be $x_1, x_2, x_3, x_4, x_5, x_6, x_7,$ and $x_8$. Given the known observations: $x_1=6, x_2=7, x_3=10, x_4=12, x_5=12, x_6=13$. Let the two unknown observations be $a$ and $b$. The total number of observations $n = 8$.

Step 2: Utilizing the Mean to find the Sum of Observations

The formula for the mean ($\bar{x}$) is $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$. [Per the definition of Arithmetic Mean]. Given $\bar{x} = 9$ and $n = 8$: $9 = \frac{6 + 7 + 10 + 12 + 12 + 13 + a + b}{8}$ $72 = 60 + a + b$ $a + b = 12$ --- (Equation 1)

Step 3: Utilizing the Variance to find the Sum of Squares

The formula for variance ($\sigma^2$) is $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. [Per the computational formula for Variance]. Given $\sigma^2 = 9.25$ and $\bar{x} = 9$: $9.25 = \frac{6^2 + 7^2 + 10^2 + 12^2 + 12^2 + 13^2 + a^2 + b^2}{8} - (9)^2$ $9.25 = \frac{36 + 49 + 100 + 144 + 144 + 169 + a^2 + b^2}{8} - 81$ $90.25 = \frac{642 + a^2 + b^2}{8}$ $722 = 642 + a^2 + b^2$ $a^2 + b^2 = 80$ --- (Equation 2)

Step 4: Solving the Simultaneous Equations

From Equation 1, $b = 12 - a$. Substitute this into Equation 2: $a^2 + (12 - a)^2 = 80$ $a^2 + 144 - 24a + a^2 = 80$ $2a^2 - 24a + 64 = 0$ Divide by 2: $a^2 - 12a + 32 = 0$ Factorizing the quadratic equation: $(a - 8)(a - 4) = 0$ Thus, $a = 8$ or $a = 4$. If $a = 8$, then $b = 12 - 8 = 4$. If $a = 4$, then $b = 12 - 4 = 8$.

Visual Representation of the Data Distribution

Final Solution: The remaining two observations are 4 and 8.

Step 1: Data Tabulation and Mean Calculation

To find the mean deviation about the mean, we first define the given frequency distribution where $x_i$ represents the variable and $f_i$ represents the corresponding frequency.

The mean ($\bar{x}$) is calculated using the formula: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ [By the definition of Arithmetic Mean for grouped data].

$\bar{x} = \frac{4000}{80} = 50$.

Step 2: Calculating Mean Deviation about the Mean

The Mean Deviation about the mean ($MD(\bar{x})$) is given by the formula: $MD(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$.

Step 3: Final Computation

Substitute the values into the Mean Deviation formula:

$MD(\bar{x}) = \frac{1280}{80}$

$MD(\bar{x}) = 16$.

Visual Representation of Data Distribution

Final Solution: The mean deviation about the mean for the given data is 16.

Given Data and Initial Frequency Distribution

The provided data consists of class intervals and their corresponding frequencies. We represent the midpoint of each class interval as $x_i$ and the frequency as $f_i$.

Step 1: Calculating the Mean ($\bar{x}$)

The mean is calculated using the formula: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ [Definition of Arithmetic Mean for Grouped Data].

Substituting the values from the table:

$\bar{x} = \frac{1370}{30} = 45.67$

Step 2: Calculating the Variance ($\sigma^2$)

The variance is determined using the formula: $\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - (\bar{x})^2$, where $N = \sum f_i$ [Computational Formula for Variance].

Substitute the calculated sums:

$\sigma^2 = \frac{70150}{30} - (45.67)^2$

$\sigma^2 = 2338.33 - 2085.75$

$\sigma^2 = 252.58$

Visual Representation of Data Distribution

Final Solution: The Mean ($\bar{x}$) is $45.67$ and the Variance ($\sigma^2$) is $252.58$.

Step 1: Data Tabulation and Mean Calculation

To find the mean deviation about the mean, we first represent the given frequency distribution. The data is provided as follows:

The mean ($\bar{x}$) is calculated using the formula: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ [By Definition of Arithmetic Mean].

$\bar{x} = \frac{398}{40} = 9.95$

Step 2: Calculating Absolute Deviations

We now calculate the absolute deviation $|x_i - \bar{x}|$ for each observation and multiply by the corresponding frequency $f_i$.

Step 3: Calculating Mean Deviation about the Mean

The Mean Deviation ($MD$) is defined as the average of the absolute deviations from the mean [Statistical Formula for M.D.]:

$MD(\bar{x}) = \frac{\sum f_i |x_i - \bar{x}|}{\sum f_i}$

$MD(\bar{x}) = \frac{74.2}{40}$

$MD(\bar{x}) = 1.855$

Final Solution: The Mean Deviation about the Mean is 1.855.

Given Data & Frequency Distribution

The provided data consists of a frequency distribution. We define the variable $x_i$ as the class mark (mid-point of the class intervals) and $f_i$ as the corresponding frequency.

Step 1: Calculating the Median ($M$)

The total frequency $N = 48$. The median class is determined by $\frac{N}{2} = \frac{48}{2} = 24$.

Looking at the cumulative frequency column, the value 24 falls within the class interval 20-30. [Per the definition of Median for grouped data].

Formula: $M = l + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h$

• $l$ (lower limit of median class) = 20
• $N/2 = 24$
• $cf$ (cumulative frequency of class preceding median class) = 13
• $f$ (frequency of median class) = 15
• $h$ (class width) = 10

$M = 20 + \left( \frac{24 - 13}{15} \right) \times 10 = 20 + \left( \frac{11}{15} \right) \times 10 = 20 + \frac{110}{15} = 20 + 7.33 = 27.33$

Step 2: Calculating Mean Deviation about Median ($MD(M)$)

The formula for Mean Deviation is $MD(M) = \frac{\sum f_i |x_i - M|}{N}$.

Step 3: Final Computation

$MD(M) = \frac{448.64}{48} \approx 9.3466$

Final Solution: The mean deviation about the median is $9.35$.

Step 1: Data Arrangement and Identification

The given data set is: $13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17$.

First, we arrange the observations in ascending order: $10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18$.

The total number of observations $n = 12$. [By definition, $n$ is even].

Step 2: Calculating the Median

For an even number of observations, the median is the arithmetic mean of the $(\frac{n}{2})^{th}$ and $(\frac{n}{2} + 1)^{th}$ terms. [Per the definition of Median for grouped/ungrouped discrete data].

• $\frac{n}{2} = \frac{12}{2} = 6^{th}$ term = $13$
• $\frac{n}{2} + 1 = 7^{th}$ term = $14$

Median ($M$) = $\frac{13 + 14}{2} = \frac{27}{2} = 13.5$.

Step 3: Calculating Absolute Deviations

We calculate the absolute deviations $|x_i - M|$ for each $x_i$, where $M = 13.5$.

Step 4: Summation of Absolute Deviations

$\sum |x_i - M| = 3.5 + 2.5 + 2.5 + 1.5 + 0.5 + 0.5 + 0.5 + 2.5 + 2.5 + 3.5 + 3.5 + 4.5$

$\sum |x_i - M| = 28$

Step 5: Final Mean Deviation Calculation

Mean Deviation about Median ($MD_M$) is given by the formula: $MD_M = \frac{\sum |x_i - M|}{n}$. [Per the statistical formula for Mean Deviation].

$MD_M = \frac{28}{12} = \frac{7}{3} \approx 2.333$

Final Solution: The mean deviation about the median is $2.33$ (or $\frac{7}{3}$).

Step 1: Organization and Calculation of the Mean ($\bar{x}$)

Given the data set: $x_i = \{6, 7, 10, 12, 13, 4, 8, 12\}$. The number of observations is $n = 8$.

The arithmetic mean ($\bar{x}$) is defined by the formula: $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$ [Definition of Arithmetic Mean].

Calculation of the sum of observations ($\sum x_i$):

$6 + 7 + 10 + 12 + 13 + 4 + 8 + 12 = 72$

Calculating the mean:

$\bar{x} = \frac{72}{8} = 9$

Step 2: Tabulation for Variance Calculation

The variance ($\sigma^2$) is defined as: $\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$ [Variance Formula].

Step 3: Calculating Variance and Standard Deviation

Sum of squared deviations: $\sum_{i=1}^{8} (x_i - 9)^2 = 74$.

Variance ($\sigma^2$):

$\sigma^2 = \frac{1}{8} \times 74$

$\sigma^2 = 9.25$

Visual Representation of Data Distribution

Final Solution: The Mean ($\bar{x}$) is $9$ and the Variance ($\sigma^2$) is $9.25$.

Step 1: Establishing the Initial Parameters

Given the initial set of $n_1 = 100$ observations:

• Mean ($\bar{x}_1$) = $20$
• Standard Deviation ($\sigma_1$) = $3$

The relationship for the sum of observations ($\sum x_i$) is defined by $\bar{x} = \frac{\sum x_i}{n}$. [Per the definition of Arithmetic Mean]

$\sum x_i = n_1 \times \bar{x}_1 = 100 \times 20 = 2000$.

The relationship for variance ($\sigma^2$) is $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. [Per the Computational Formula for Variance]

$\sigma_1^2 = 3^2 = 9$. Thus, $9 = \frac{\sum x_i^2}{100} - (20)^2$.

$\sum x_i^2 = 100 \times (9 + 400) = 40900$.

Step 2: Removing Incorrect Observations

The incorrect observations to be removed are $21, 21, 18$. The number of remaining observations is $n_2 = 100 - 3 = 97$.

New sum of observations ($\sum x_{new}$):

$\sum x_{new} = 2000 - (21 + 21 + 18) = 2000 - 60 = 1940$.

New sum of squares ($\sum x_{new}^2$):

$\sum x_{new}^2 = 40900 - (21^2 + 21^2 + 18^2) = 40900 - (441 + 441 + 324) = 40900 - 1206 = 39694$.

Step 3: Calculating New Mean and Standard Deviation

New Mean ($\bar{x}_2$):

$\bar{x}_2 = \frac{\sum x_{new}}{n_2} = \frac{1940}{97} = 20$.

New Variance ($\sigma_2^2$):

$\sigma_2^2 = \frac{\sum x_{new}^2}{n_2} - (\bar{x}_2)^2$

$\sigma_2^2 = \frac{39694}{97} - (20)^2 \approx 409.2165 - 400 = 9.2165$.

New Standard Deviation ($\sigma_2$):

$\sigma_2 = \sqrt{9.2165} \approx 3.036$.

Visual Representation of Data Distribution (Conceptual)

Final Solution: The new mean is $20$ and the new standard deviation is approximately $3.036$.

Step 1: Data Tabulation and Initial Setup

To calculate the mean and variance for the given frequency distribution, we first construct a frequency table using the class intervals ($x_i$ represent the midpoints of the classes).

Step 2: Calculating the Mean ($\bar{x}$ )

The mean of a grouped frequency distribution is calculated by the formula [Per the definition of arithmetic mean for grouped data]:

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$

Substituting the values from the table:

$\bar{x} = \frac{3210}{30} = 107$

Step 3: Calculating the Variance ($\sigma^2$)

The variance is determined using the formula [By the computational formula for variance]:

$\sigma^2 = \frac{1}{N} \sum f_i x_i^2 - (\bar{x})^2$, where $N = \sum f_i$.

$\sigma^2 = \frac{411775}{30} - (107)^2$

$\sigma^2 = 13725.833 - 11449$

$\sigma^2 = 2276.833$

Step 4: Statistical Visualization

Final Solution: The Mean is $107$ and the Variance is $2276.833$.

Step 1: Data Tabulation and Selection of Assumed Mean

Given the frequency distribution, we first identify the mid-points ($x_i$) and select an assumed mean ($A$). The data is structured as follows:

Let the assumed mean $A = 72$ [Choosing the mid-point of the central class with the highest frequency for computational efficiency].

Step 2: Calculating the Mean ($\bar{x}$)

Using the short-cut formula for the mean: $\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i}$ [By the Property of Arithmetic Mean under linear transformation].

$\bar{x} = 72 + \frac{90}{30}$

$\bar{x} = 72 + 3 = 75$

Step 3: Calculating the Standard Deviation ($\sigma$)

Using the formula for standard deviation by short-cut method: $\sigma = \sqrt{\frac{\sum f_i d_i^2}{N} - \left( \frac{\sum f_i d_i}{N} \right)^2}$ [By Variance identity $\text{Var}(X) = E[X^2] - (E[X])^2$].

Substitute the values from the table where $N = \sum f_i = 30$:

$\sigma = \sqrt{\frac{1400}{30} - \left( \frac{90}{30} \right)^2}$

$\sigma = \sqrt{46.6667 - (3)^2}$

$\sigma = \sqrt{46.6667 - 9}$

$\sigma = \sqrt{37.6667} \approx 6.137$

Visual Representation of Data Distribution

Final Solution: Mean ($\bar{x}$) = 75; Standard Deviation ($\sigma$) $\approx$ 6.137

Step 1: Calculating the Arithmetic Mean ($\bar{x}$)

Given data set: $x_i = \{38, 70, 48, 40, 42, 55, 63, 46, 54, 44\}$

Number of observations ($n$) = $10$.

The arithmetic mean is defined as $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$. [Definition of Arithmetic Mean for Ungrouped Data]

Summation: $38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44 = 500$.

$\bar{x} = \frac{500}{10} = 50$.

Step 2: Calculating Absolute Deviations $|x_i - \bar{x}|$

We calculate $|x_i - 50|$ for each observation to find the distance from the mean. [Property of Absolute Deviation]

Step 3: Calculating Mean Deviation about the Mean (M.D.($\bar{x}$))

The formula for Mean Deviation about the mean is: $\text{M.D.}(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$. [Statistical Definition of Mean Deviation]

Sum of absolute deviations = $12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6 = 84$.

$\text{M.D.}(\bar{x}) = \frac{84}{10} = 8.4$.

Visualization of Data Distribution

Final Solution: The mean deviation about the mean for the given data is 8.4.

Step 1: Establishing the Data Set and Mean Calculation

The set of the first $n$ natural numbers is defined as $S = \{1, 2, 3, \dots, n\}$.

The mean ($\bar{x}$) is defined as the sum of all observations divided by the number of observations ($n$):

$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} i$

Using the arithmetic progression summation formula [$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$]:

$\bar{x} = \frac{1}{n} \cdot \frac{n(n+1)}{2} = \frac{n+1}{2}$

Step 2: Defining the Variance Formula

The variance ($\sigma^2$) of a discrete data set is given by the formula:

$\sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$

Alternatively, using the computational formula [By algebraic expansion]:

$\sigma^2 = \left( \frac{1}{n} \sum_{i=1}^{n} x_i^2 \right) - (\bar{x})^2$

Step 3: Calculating the Sum of Squares

We require the sum of the squares of the first $n$ natural numbers [Standard Power Sum Formula]:

$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Step 4: Algebraic Derivation of Variance

Substituting the values from Step 1 and Step 3 into the computational variance formula:

$\sigma^2 = \frac{1}{n} \left[ \frac{n(n+1)(2n+1)}{6} \right] - \left( \frac{n+1}{2} \right)^2$

$\sigma^2 = \frac{(n+1)(2n+1)}{6} - \frac{(n+1)^2}{4}$

Factoring out $\frac{n+1}{2}$:

$\sigma^2 = \frac{n+1}{2} \left[ \frac{2n+1}{3} - \frac{n+1}{2} \right]$

Finding the common denominator within the brackets:

$\sigma^2 = \frac{n+1}{2} \left[ \frac{2(2n+1) - 3(n+1)}{6} \right]$

$\sigma^2 = \frac{n+1}{2} \left[ \frac{4n + 2 - 3n - 3}{6} \right]$

$\sigma^2 = \frac{n+1}{2} \left[ \frac{n - 1}{6} \right]$

$\sigma^2 = \frac{(n+1)(n-1)}{12} = \frac{n^2 - 1}{12}$

Visual Representation of Distribution

Final Solution: The mean is $\frac{n+1}{2}$ and the variance is $\frac{n^2-1}{12}$.

Step 1: Identification of the Dataset

The first 10 multiples of 3 form an arithmetic progression. Let the dataset be denoted by $X = \{x_1, x_2, ..., x_{10}\}$.

The values are: $3, 6, 9, 12, 15, 18, 21, 24, 27, 30$.

Here, the number of observations $n = 10$.

Step 2: Calculation of the Mean ($\bar{x}$)

The mean is defined as the sum of observations divided by the number of observations [By Definition of Arithmetic Mean]:

$\bar{x} = \frac{\sum_{i=1}^{10} x_i}{n} = \frac{3+6+9+12+15+18+21+24+27+30}{10}$

Using the formula for the sum of an arithmetic series $S_n = \frac{n}{2}(a + l)$, where $a=3$ and $l=30$:

$\sum x_i = \frac{10}{2}(3 + 30) = 5 \times 33 = 165$

$\bar{x} = \frac{165}{10} = 16.5$

Step 3: Calculation of Variance ($\sigma^2$)

The variance is calculated using the formula [Per the Statistical Definition of Variance]:

$\sigma^2 = \frac{1}{n} \sum_{i=1}^{10} (x_i - \bar{x})^2$

Alternatively, the computational formula is $\sigma^2 = \frac{1}{n} \sum x_i^2 - (\bar{x})^2$.

Now, calculate the mean of the squares:

$\frac{\sum x_i^2}{n} = \frac{3465}{10} = 346.5$

$\sigma^2 = 346.5 - (16.5)^2$

Calculate $(16.5)^2 = 272.25$

$\sigma^2 = 346.5 - 272.25 = 74.25$

Step 4: Visual Representation of Data Spread

Final Solution: The mean of the first 10 multiples of 3 is $16.5$, and the variance is $74.25$.

Step 1: Organizing and Ordering the Data

To determine the median, the given raw data must be arranged in ascending order. Let the given set be $X = \{36, 72, 46, 42, 60, 45, 53, 46, 51, 49\}$.

Ordered sequence: $36, 42, 45, 46, 46, 49, 51, 53, 60, 72$

The total number of observations, $n = 10$. [Since $n$ is even, we use the median formula for even-numbered datasets].

Step 2: Calculating the Median ($M$)

The median of an even set of observations is the average of the $(\frac{n}{2})^{th}$ and $(\frac{n}{2} + 1)^{th}$ terms.

• Term 1: $\frac{10}{2} = 5^{th}$ term = $46$
• Term 2: $5 + 1 = 6^{th}$ term = $49$

$M = \frac{46 + 49}{2} = \frac{95}{2} = 47.5$

Step 3: Calculating Absolute Deviations $|x_i - M|$

We now compute the absolute deviation of each observation $x_i$ from the median $M = 47.5$.

Step 4: Summation of Absolute Deviations

Summing the values calculated in Step 3:

$\sum_{i=1}^{10} |x_i - M| = 11.5 + 5.5 + 2.5 + 1.5 + 1.5 + 1.5 + 3.5 + 5.5 + 12.5 + 24.5$

$\sum_{i=1}^{10} |x_i - M| = 70.0$

Step 5: Final Calculation of Mean Deviation

The Mean Deviation about the Median ($MD_M$) is defined by the formula: [Per the statistical definition of mean deviation]

$MD_M = \frac{\sum_{i=1}^{n} |x_i - M|}{n}$

$MD_M = \frac{70}{10} = 7$

Final Solution: The Mean Deviation about the median is 7.

Given Data and Initial Frequency Distribution

The provided data consists of class intervals and their corresponding frequencies ($f_i$). We represent the data in tabular form to facilitate calculations:

Step 1: Calculating the Mean ($\bar{x}$)

The formula for the mean of grouped data is given by: $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$ [Definition of Arithmetic Mean for Grouped Data]

Substituting the values from the table:

Step 2: Calculating the Variance ($\sigma^2$)

The formula for variance is: $\sigma^2 = \frac{1}{N} \sum f_i(x_i - \bar{x})^2$ or equivalently, $\sigma^2 = \frac{\sum f_i x_i^2}{N} - (\bar{x})^2$ [Statistical Formula for Variance]

Using the values $\sum f_i x_i^2 = 126044$, $N = 91$, and $\bar{x} = 36.24175...$:

Performing the division:

Calculating the square of the mean:

Final subtraction:

Visual Representation of Data Distribution

Final Solution: The mean ($\bar{x}$) is approximately $36.24$ and the variance ($\sigma^2$) is approximately $71.63$.

Step 1: Data Tabulation and Mid-point Calculation

To find the mean and variance for a grouped frequency distribution, we first identify the mid-point ($x_i$) for each class interval using the formula $x_i = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$.

Step 2: Calculation of Mean ($\bar{x}$ )

The mean is calculated using the formula: $\bar{x} = \frac{\sum f_ix_i}{N}$ [Per the definition of arithmetic mean for grouped data].

$\bar{x} = \frac{1350}{50} = 27$

Step 3: Calculation of Variance ($\sigma^2$)

The variance for a grouped frequency distribution is calculated using the formula: $\sigma^2 = \frac{1}{N} \sum f_i(x_i - \bar{x})^2$ or the computational formula $\sigma^2 = \frac{1}{N} \sum f_ix_i^2 - (\bar{x})^2$.

Substituting the calculated values:

$\sigma^2 = \frac{43050}{50} - (27)^2$ [By the Algebraic Variance Identity]

$\sigma^2 = 861 - 729$

$\sigma^2 = 132$

Visual Representation of Frequency Distribution

Final Solution: The Mean ($\bar{x}$) is $27$ and the Variance ($\sigma^2$) is $132$.

Step 1: Establishing the Frequency Distribution Table

To find the mean deviation about the median, we first arrange the given data and calculate the cumulative frequencies ($cf$).

Step 2: Determining the Median

The total number of observations is $N = 29$, which is odd. The median is the value of the $\left(\frac{N+1}{2}\right)^{th}$ observation.

Position $= \frac{29+1}{2} = 15^{th}$ observation. [Per the definition of median for discrete frequency distributions]

Looking at the cumulative frequency column, the $15^{th}$ observation lies in the row where $cf = 21$. Thus, the Median ($M$) = 30.

Step 3: Calculating Deviations and Mean Deviation

We calculate the absolute deviation $|x_i - M|$ for each observation and multiply by its corresponding frequency $f_i$.

Step 4: Final Calculation

The formula for Mean Deviation about the Median ($MD_{med}$) is given by: [As per statistical measure of dispersion formulas]

$MD_{med} = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - M|$

$MD_{med} = \frac{148}{29}$

$MD_{med} \approx 5.103$

Final Solution: The mean deviation about the median is $\frac{148}{29} \approx 5.103$.

Step 1: Constructing the Frequency Distribution Table

To find the mean deviation about the median, we first organize the data to calculate the cumulative frequency ($cf$).

Step 2: Locating the Median ($M$)

The total number of observations $N = 26$ (which is even). The median is the mean of the $(\frac{N}{2})^{th}$ and $(\frac{N}{2} + 1)^{th}$ observation.

$\frac{N}{2} = \frac{26}{2} = 13$

$\frac{N}{2} + 1 = 14$

Looking at the cumulative frequency table, the $13^{th}$ observation corresponds to $x = 7$ (since $8 < 13 \le 14$), and the $14^{th}$ observation corresponds to $x = 7$.

$M = \frac{7 + 7}{2} = 7$ [By definition of median for discrete frequency distribution]

Step 3: Calculating Absolute Deviations and Frequency Products

We calculate $|x_i - M|$ for each $x_i$ and multiply by the corresponding frequency $f_i$.

Step 4: Calculation of Mean Deviation about Median

The formula for Mean Deviation ($M.D.$) about the median is:

$M.D.(M) = \frac{1}{N} \sum_{i=1}^{n} f_i |x_i - M|$

$M.D.(M) = \frac{84}{26}$

$M.D.(M) = \frac{42}{13} \approx 3.23$

Visual Representation of Data Distribution

Final Solution: The mean deviation about the median is $\frac{42}{13}$ or approximately $3.23$.

Step 1: Establishing Initial Parameters

Given six observations, let them be denoted by $x_1, x_2, x_3, x_4, x_5, x_6$. The problem provides the initial mean ($\bar{x}$) and the initial standard deviation ($\sigma$):

• Number of observations ($n$) = $6$
• Initial Mean ($\bar{x}$) = $8$
• Initial Standard Deviation ($\sigma$) = $4$

Step 2: Transformation Rule for the Mean

If each observation $x_i$ is multiplied by a constant $k$, the new observations $y_i$ are defined as $y_i = kx_i$. Per the linearity property of the arithmetic mean: $\bar{y} = \frac{1}{n} \sum y_i = \frac{1}{n} \sum (kx_i) = k \left( \frac{1}{n} \sum x_i \right) = k\bar{x}$.

Given $k = 3$ and $\bar{x} = 8$:

$\bar{y} = 3 \times 8 = 24$

Step 3: Transformation Rule for Standard Deviation

The standard deviation is defined as $\sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}}$. For the new set of observations $y_i = kx_i$, the new mean is $\bar{y} = k\bar{x}$. The new standard deviation ($\sigma'$) is:

$\sigma' = \sqrt{\frac{\sum (y_i - \bar{y})^2}{n}}$

Substituting $y_i = kx_i$ and $\bar{y} = k\bar{x}$:

$\sigma' = \sqrt{\frac{\sum (kx_i - k\bar{x})^2}{n}} = \sqrt{\frac{\sum k^2(x_i - \bar{x})^2}{n}}$

Extracting the constant $k^2$ from the summation [Per the properties of variance]:

$\sigma' = \sqrt{k^2 \cdot \frac{\sum (x_i - \bar{x})^2}{n}} = |k| \sigma$

Step 4: Calculation of New Values

Using $k = 3$ and the initial standard deviation $\sigma = 4$:

New Standard Deviation ($\sigma'$) = $|3| \times 4 = 12$

Final Solution: The new mean is 24 and the new standard deviation is 12.

Step 1: Calculating the Arithmetic Mean ($\bar{x}$)

Given data set: $x_i = \{4, 7, 8, 9, 10, 12, 13, 17\}$. The number of observations $n = 8$.

The arithmetic mean is defined as $\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$.

Summation: $\sum x_i = 4 + 7 + 8 + 9 + 10 + 12 + 13 + 17 = 80$.

Calculation: $\bar{x} = \frac{80}{8} = 10$. [By the definition of Arithmetic Mean for ungrouped data].

Step 2: Calculating Absolute Deviations from the Mean $|x_i - \bar{x}|$

We compute the absolute deviation for each observation $x_i$ from $\bar{x} = 10$.

Step 3: Calculating Mean Deviation about the Mean ($M.D.(\bar{x})$)

The formula for Mean Deviation is $M.D.(\bar{x}) = \frac{1}{n} \sum_{i=1}^{n} |x_i - \bar{x}|$. [By the definition of Mean Deviation].

Sum of absolute deviations: $\sum |x_i - \bar{x}| = 6 + 3 + 2 + 1 + 0 + 2 + 3 + 7 = 24$.

Calculation: $M.D.(\bar{x}) = \frac{24}{8} = 3$.

Visual Representation of Data Distribution

Final Solution: The Mean Deviation about the mean for the given data is 3.