Statistics .

From basic averages to complex relational models. Discover the parameters that define data.

Measures of Central Tendency

Core Definitions

Arithmetic Mean:

The sum of all observations divided by the total number of observations.

\bar{x} = \frac{\sum x_i}{n}

Median:

The middle value of a data set when arranged in ascending or descending order. For grouped data,

M = l + (\frac{n/2 - cf}{f}) \times h

Mode:

The value that appears most frequently in a data set.

Mean Formulas

ℹ️

Ungrouped Data

\bar{x} = \frac{\sum x_i}{n}

💡

Grouped Data

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

⚠️

Shortcut Method

\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \text{ where } d_i = x_i - A

Dynamic Statistics Lab

Adjust values to see how Mean, Median, and Dispersion change

Input Dataset (Max 100)

Arithmetic Mean

25.00

Median

Standard Deviation

10.00

Mode

N/A

Mean: 25.0

Notice how outliers (extremely high or low values) significantly pull the Mean, while the Median remains relatively stable. This is why Median is often preferred for skewed data like income or house prices.

Measures of Dispersion

Mean Deviation

Frequency Dist. MD

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

Variance & SD

Variance (Grouped)

\sigma^2 = \frac{\sum f_i(x_i - \bar{x})^2}{N}

Standard Deviation (SD)

\sigma = \sqrt{\frac{\sum f_i(x_i - \bar{x})^2}{N}}

Shortcut Method (Assumed Mean)

When calculations are cumbersome, we use an **Assumed Mean (A)**. Let

d_i = x_i - A

Shortcut Mean

\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \text{ where } d_i = x_i - A

Shortcut Variance

\sigma^2 = \frac{\sum f_i d_i^2}{N} - \left(\frac{\sum f_i d_i}{N}\right)^2 \text{ where } d_i = x_i - A

This method is essential for larger datasets where

(x_i - \bar{x})^2

becomes too large to compute manually.

Advanced Measures: Shape of Data

Moments

Raw Moment

\mu'_r = \frac{\sum (x_i - A)^r}{n} \text{ (about point A)}

Central Moment

\mu_r = \frac{\sum (x_i - \bar{x})^r}{n} \text{ (about mean)}

Skewness ($S_k$)

Karl Pearson

S_k = \frac{\text{Mean} - \text{Mode}}{\sigma} \approx \frac{3(\text{Mean} - \text{Median})}{\sigma}

Moment Coefficient ($\gamma_1$)

\gamma_1 = \sqrt{\beta_1} = \frac{\mu_3}{\sigma^3} = \frac{\mu_3}{(\mu_2)^{3/2}}

MEASURE ($\beta_1$):

\beta_1 = \frac{\mu_3^2}{\mu_2^3} \text{ (Measure \text{ of } Skewness)}

Kurtosis ($\\beta_2$)

Leptokurtic

\beta_2 > 3

(Highly Peaked)

Mesokurtic

\beta_2 = 3

(Normal)

Platykurtic

\beta_2 < 3

(Flat Top)

Advanced Stats Lab

Visualize Distribution Shapes & Relational Patterns

Skewness: Zero

Left SkewSymmetricRight Skew

Kurtosis Peakedness: Meso

Flat (Platy)Normal (Meso)Peaked (Lepto)

Skewness describes the asymmetry of the curve. Kurtosis describes how "peaky" it is. A Normal distribution has $S_k = 0$ and $\beta_2 = 3$.

Correlation Analysis

Karl Pearson's Coefficient ($r$)

A mathematical method to measure the degree of linear relationship between two variables.

r = \frac{Cov(X,Y)}{\sigma_x \sigma_y} = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}

Key Properties

Ranges from **-1.0 to +1.0**
Independent of change of Origin and Scale
$r=0$ indicates **No Linear Correlation**

Covariance Calculation

Expanded Form

Cov(X,Y) = \frac{1}{n} \left( \sum xy - \frac{1}{n} \sum x \sum y \right)

This form shows that

Cov(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]

Formula Clarity

In textbooks, you might see:

\frac{1}{n}\sum xy - \bar{x}\bar{y}

This is algebraically derived by expanding

\sum (x_i-\bar{x})(y_i-\bar{y})

and is the standard way to calculate correlation manually.

✏️ Practice Zone

Problems

Calculate the Mean Deviation about the mean for the data: 6, 7, 10, 12, 13, 4, 8, 12.

✓ Show Solution

1. Mean

\bar{x} = \frac{6+7+10+12+13+4+8+12}{8} = 9

|x_i - \bar{x}|

: 3, 2, 1, 3, 4, 5, 1, 3

3. Sum = 22

MD(\bar{x}) = 22/8 = 2.75

EASY

The first four raw moments of a distribution about the point 4 are 1, 4, 10, and 45. Find the mean.

✓ Show Solution

1. Given

A = 4

and

\mu'_1 = 1

2. Mean

\bar{x} = A + \mu'_1 = 4 + 1 = 5

MEDIUM

In a frequency distribution, the third central moment

\mu_3 = -2.5

and variance

\sigma^2 = 9

. Find the coefficient of skewness based on moments.

✓ Show Solution

1. Coefficient of Skewness

\gamma_1 = \frac{\mu_3}{\sigma^3}

\sigma = \sqrt{9} = 3

\gamma_1 = \frac{-2.5}{3^3} = \frac{-2.5}{27} \approx -0.09

MEDIUM

Calculate Spearman's Rank Correlation for the following ranks: (1,3), (2,1), (3,2).

✓ Show Solution

d_i

: (1-3)=-2, (2-1)=1, (3-2)=1

d_i^2

: 4, 1, 1.

\sum d_i^2 = 6

R = 1 - \frac{6(6)}{3(3^2-1)} = 1 - \frac{36}{3(8)} = 1 - \frac{36}{24} = 1 - 1.5 = -0.5

HARD

\mu_4 = 100

and

\mu_2 = 5

, determine the type of kurtosis.

✓ Show Solution

\beta_2 = \mu_4 / \mu_2^2 = 100 / 5^2 = 100 / 25 = 4

2. Since

\beta_2 > 3

, the distribution is **Leptokurtic**.

MEDIUM

CBSE Exam Corner

Master these advanced statistical proofs and calculations.

CBSE PYQID: stat_exam_1

Find the mean, variance and standard deviation for: 70, 72, 74, 76, 78, 80, 82.

Calculated Procedure1.

A = 76

d_i

: -6, -4, -2, 0, 2, 4, 6 2.

\bar{x} = 76 + 0 = 76

\sigma^2 = \frac{36+16+4+0+4+16+36}{7} = 112/7 = 16

\sigma = 4

ConceptualID: stat_exam_corr

Prove that Karl Pearson's coefficient of correlation lies between -1 and +1.

Calculated Procedure1. Based on Cauchy-Schwarz inequality:

[\sum (x_i-\bar{x})(y_i-\bar{y})]^2 \le \sum (x_i-\bar{x})^2 \sum (y_i-\bar{y})^2

2. Dividing both sides by denom of

r^2

gives

r^2 \le 1

3. Thus,

|r \mid \le 1

-1 \le r \le 1

CBSE 2022ID: stat_exam_skew

If the difference between Mean and Mode is 48 and standard deviation is 20, find the coefficient of skewness.

Calculated Procedure1.

S_k = (\text{Mean} - \text{Mode}) / \sigma

S_k = 48 / 20 = 2.4

3. This indicates a high positive skewness.

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Statistics .

From basic averages to complex relational models. Discover the parameters that define data.

Measures of Central Tendency

Core Definitions

Arithmetic Mean:

The sum of all observations divided by the total number of observations.

\bar{x} = \frac{\sum x_i}{n}

Median:

The middle value of a data set when arranged in ascending or descending order. For grouped data,

M = l + (\frac{n/2 - cf}{f}) \times h

Mode:

The value that appears most frequently in a data set.

Mean Formulas

ℹ️

Ungrouped Data

\bar{x} = \frac{\sum x_i}{n}

💡

Grouped Data

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

⚠️

Shortcut Method

\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \text{ where } d_i = x_i - A

Dynamic Statistics Lab

Adjust values to see how Mean, Median, and Dispersion change

Input Dataset (Max 100)

Arithmetic Mean

25.00

Median

Standard Deviation

10.00

Mode

N/A

Mean: 25.0

Measures of Dispersion

Mean Deviation

Frequency Dist. MD

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

Variance & SD

Variance (Grouped)

\sigma^2 = \frac{\sum f_i(x_i - \bar{x})^2}{N}

Standard Deviation (SD)

\sigma = \sqrt{\frac{\sum f_i(x_i - \bar{x})^2}{N}}

Shortcut Method (Assumed Mean)

When calculations are cumbersome, we use an **Assumed Mean (A)**. Let

d_i = x_i - A

Shortcut Mean

\bar{x} = A + \frac{\sum f_i d_i}{\sum f_i} \text{ where } d_i = x_i - A

Shortcut Variance

\sigma^2 = \frac{\sum f_i d_i^2}{N} - \left(\frac{\sum f_i d_i}{N}\right)^2 \text{ where } d_i = x_i - A

This method is essential for larger datasets where

(x_i - \bar{x})^2

becomes too large to compute manually.

Advanced Measures: Shape of Data

Moments

Raw Moment

\mu'_r = \frac{\sum (x_i - A)^r}{n} \text{ (about point A)}

Central Moment

\mu_r = \frac{\sum (x_i - \bar{x})^r}{n} \text{ (about mean)}

Skewness ($S_k$)

Karl Pearson

S_k = \frac{\text{Mean} - \text{Mode}}{\sigma} \approx \frac{3(\text{Mean} - \text{Median})}{\sigma}

Moment Coefficient ($\gamma_1$)

\gamma_1 = \sqrt{\beta_1} = \frac{\mu_3}{\sigma^3} = \frac{\mu_3}{(\mu_2)^{3/2}}

MEASURE ($\beta_1$):

\beta_1 = \frac{\mu_3^2}{\mu_2^3} \text{ (Measure \text{ of } Skewness)}

Kurtosis ($\\beta_2$)

Leptokurtic

\beta_2 > 3

(Highly Peaked)

Mesokurtic

\beta_2 = 3

(Normal)

Platykurtic

\beta_2 < 3

(Flat Top)

Advanced Stats Lab

Visualize Distribution Shapes & Relational Patterns

Skewness: Zero

Left SkewSymmetricRight Skew

Kurtosis Peakedness: Meso

Flat (Platy)Normal (Meso)Peaked (Lepto)

Skewness describes the asymmetry of the curve. Kurtosis describes how "peaky" it is. A Normal distribution has $S_k = 0$ and $\beta_2 = 3$.

Correlation Analysis

Karl Pearson's Coefficient ($r$)

A mathematical method to measure the degree of linear relationship between two variables.

r = \frac{Cov(X,Y)}{\sigma_x \sigma_y} = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}

Key Properties

Ranges from **-1.0 to +1.0**
Independent of change of Origin and Scale
$r=0$ indicates **No Linear Correlation**

Covariance Calculation

Expanded Form

Cov(X,Y) = \frac{1}{n} \left( \sum xy - \frac{1}{n} \sum x \sum y \right)

This form shows that

Cov(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X]\mathbb{E}[Y]

Formula Clarity

In textbooks, you might see:

\frac{1}{n}\sum xy - \bar{x}\bar{y}

This is algebraically derived by expanding

\sum (x_i-\bar{x})(y_i-\bar{y})

and is the standard way to calculate correlation manually.

✏️ Practice Zone

Problems

Calculate the Mean Deviation about the mean for the data: 6, 7, 10, 12, 13, 4, 8, 12.

✓ Show Solution

1. Mean

\bar{x} = \frac{6+7+10+12+13+4+8+12}{8} = 9

|x_i - \bar{x}|

: 3, 2, 1, 3, 4, 5, 1, 3

3. Sum = 22

MD(\bar{x}) = 22/8 = 2.75

EASY

The first four raw moments of a distribution about the point 4 are 1, 4, 10, and 45. Find the mean.

✓ Show Solution

1. Given

A = 4

and

\mu'_1 = 1

2. Mean

\bar{x} = A + \mu'_1 = 4 + 1 = 5

MEDIUM

In a frequency distribution, the third central moment

\mu_3 = -2.5

and variance

\sigma^2 = 9

. Find the coefficient of skewness based on moments.

✓ Show Solution

1. Coefficient of Skewness

\gamma_1 = \frac{\mu_3}{\sigma^3}

\sigma = \sqrt{9} = 3

\gamma_1 = \frac{-2.5}{3^3} = \frac{-2.5}{27} \approx -0.09

MEDIUM

Calculate Spearman's Rank Correlation for the following ranks: (1,3), (2,1), (3,2).

✓ Show Solution

d_i

: (1-3)=-2, (2-1)=1, (3-2)=1

d_i^2

: 4, 1, 1.

\sum d_i^2 = 6

R = 1 - \frac{6(6)}{3(3^2-1)} = 1 - \frac{36}{3(8)} = 1 - \frac{36}{24} = 1 - 1.5 = -0.5

HARD

\mu_4 = 100

and

\mu_2 = 5

, determine the type of kurtosis.

✓ Show Solution

\beta_2 = \mu_4 / \mu_2^2 = 100 / 5^2 = 100 / 25 = 4

2. Since

\beta_2 > 3

, the distribution is **Leptokurtic**.

MEDIUM

CBSE Exam Corner

Master these advanced statistical proofs and calculations.

CBSE PYQID: stat_exam_1

Find the mean, variance and standard deviation for: 70, 72, 74, 76, 78, 80, 82.

Calculated Procedure1.

A = 76

d_i

: -6, -4, -2, 0, 2, 4, 6 2.

\bar{x} = 76 + 0 = 76

\sigma^2 = \frac{36+16+4+0+4+16+36}{7} = 112/7 = 16

\sigma = 4

ConceptualID: stat_exam_corr

Prove that Karl Pearson's coefficient of correlation lies between -1 and +1.

Calculated Procedure1. Based on Cauchy-Schwarz inequality:

[\sum (x_i-\bar{x})(y_i-\bar{y})]^2 \le \sum (x_i-\bar{x})^2 \sum (y_i-\bar{y})^2

2. Dividing both sides by denom of

r^2

gives

r^2 \le 1

3. Thus,

|r \mid \le 1

-1 \le r \le 1

CBSE 2022ID: stat_exam_skew

If the difference between Mean and Mode is 48 and standard deviation is 20, find the coefficient of skewness.

Calculated Procedure1.

S_k = (\text{Mean} - \text{Mode}) / \sigma

S_k = 48 / 20 = 2.4

3. This indicates a high positive skewness.