Prepare Effectively With Handwritten Statistics Notes Tailored For JEE Mains, NIMCET, CUET MCA And CET MAH MCA Exams. Covers Complete Theory, Formulas, Properties And Solved Questions For Fast And Accurate Learning.

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These statistics notes are created in a simple handwritten style to help students understand concepts clearly and quickly. They are especially useful for those preparing for competitive exams and looking for jee mains notes, nimcet notes, cuet mca notes, and cet mah mca notes. All topics are explained with easy explanations, formulas, properties, and solved examples, making these notes perfect for strong conceptual clarity as well as quick revision.

1. Meaning of Statistics

Statistics deals with collecting, organizing, presenting, analyzing and interpreting numerical data.

2. Types of Data

Primary Data: First-hand information.
Secondary Data: Already available data.

3. Measures of Central Tendency

(A) Arithmetic Mean

Ungrouped data:
$$\bar{x} = \frac{\sum x_i}{n}$$

Discrete frequency:
$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

Continuous frequency:
$$\bar{x} = \frac{\sum f_i m_i}{\sum f_i}$$ where $m_i$ are midpoints.

Properties of Mean

1. Unique value.
2. $\sum (x_i - \bar{x}) = 0$.
3. Highly affected by extreme values.
4. Combined mean: $$ \bar{x} = \frac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1+n_2} $$ 5. Adding $k$ → mean becomes $\bar{x}+k$.

(B) Median

Median = middle value after arranging data.

Ungrouped:
If $n$ odd: $\text{Median} = x_{\frac{n+1}{2}}$
If $n$ even: $\text{Median} = \frac{x_{\frac{n}{2}} + x_{\frac{n}{2}+1}}{2}$

Continuous:
$$ \text{Median} = l + \left( \frac{\frac{N}{2} - c_f}{f} \right) h $$

Properties of Median

1. Not affected by extreme values.
2. Divides data into two equal halves.
3. Suitable for open-end classes.
4. Positional measure.

(C) Mode

Mode = most frequent value.

Continuous data:
$$ \text{Mode} = l + \frac{(f_1 - f_0)}{2f_1 - f_0 - f_2} h $$

Properties of Mode

1. Not affected by outliers.
2. Applicable to qualitative data.
3. Empirical relation: $$ \text{Mode} = 3\text{Median} - 2\text{Mean} $$

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4. Measures of Dispersion

(A) Range

$$\text{Range} = \text{Maximum} - \text{Minimum}$$

(B) Mean Deviation

$$ MD = \frac{\sum |x_i - \bar{x}|}{n} $$

(C) Variance & Standard Deviation

Variance:
$$ \sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n} $$

Standard deviation:
$$ \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} $$

Shortcut:
$$ \sigma = \sqrt{\frac{\sum x_i^2}{n} - \bar{x}^2} $$

Properties

1. $\sigma \ge 0$.
2. SD = 0 when all values equal.
3. Adding constant $k$ → SD unchanged.
4. Multiplying by $k$ → SD becomes $k\sigma$.
5. SD is affected by extreme values.

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5. Coefficient of Variation (C.V.)

Used to compare variability:
$$ CV = \frac{\sigma}{\bar{x}} \times 100 $$

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6. Skewness

Skewness measures asymmetry of data.

Karl Pearson coefficient:
$$ \text{Skewness} = \frac{\bar{x} - \text{Mode}}{\sigma} $$

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7. Correlation

(A) Karl Pearson Correlation Coefficient

$$ r = \frac{\sum (x-\bar{x})(y-\bar{y})} {\sqrt{\sum (x-\bar{x})^2 \sum (y-\bar{y})^2}} $$

Shortcut:
$$ r = \frac{ n\sum xy - (\sum x)(\sum y) }{ \sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)} } $$

Properties of Correlation

1. $-1 \le r \le +1$.
2. $r=0$ means no linear relation.
3. Independent variables → $r=0$.
4. Not affected by change of origin or scale.
5. If $r=\pm 1$, points lie on a straight line.

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8. Regression

Regression line of $y$ on $x$:
$$ y - \bar{y} = b_{yx}(x - \bar{x}) $$

Regression coefficient:
$$ b_{yx} = r\frac{\sigma_y}{\sigma_x} $$

Properties

1. Regression lines intersect at $(\bar{x},\bar{y})$.
2. Both regression coefficients have same sign as $r$.
3. Product: $$ b_{yx} \cdot b_{xy} = r^2 $$ 4. If $r=0$ → regression lines are perpendicular.

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9. Moments

Moments measure the shape characteristics of a distribution such as skewness and kurtosis.

(A) Raw Moments (about origin)

Raw moment of order $r$:
$$ \mu_r^\prime = \frac{1}{n}\sum x_i^r $$

Examples:
1. First raw moment: $ \mu_1^\prime = \bar{x} $
2. Second raw moment: $ \mu_2^\prime = \frac{1}{n}\sum x_i^2 $

(B) Central Moments (about mean)

Central moment of order $r$:
$$ \mu_r = \frac{1}{n}\sum (x_i - \bar{x})^r $$

Important Central Moments

1. First central moment: $$\mu_1 = 0$$ 2. Second central moment = Variance: $$\mu_2 = \sigma^2$$ 3. Third central moment → Skewness.
4. Fourth central moment → Kurtosis.

(C) Relation Between Raw & Central Moments

For second moment:
$$\mu_2 = \mu_2^\prime - (\mu_1^\prime)^2$$

General relation:
$$ \mu_r=\sum_{k=0}^{r} {r \choose k}(-1)^{r-k}(\mu_1^\prime)^{r-k}\mu_k^\prime $$

(D) Skewness Using Moments

Moment-based skewness:
$$ \beta_1 = \frac{\mu_3^2}{\mu_2^3} $$ Pearson skewness:
$$ \gamma_1 = \frac{\mu_3}{\mu_2^{3/2}} $$

(E) Kurtosis Using Moments

$$ \beta_2 = \frac{\mu_4}{\mu_2^2} $$

Interpretation:
$\beta_2 = 3$ → Mesokurtic (Normal)
$\beta_2 > 3$ → Leptokurtic (Peaked)
$\beta_2 < 3$ → Platykurtic (Flat)

Properties of Moments

1. First central moment is always 0.
2. Second central moment = variance.
3. Third central moment measures skewness.
4. Fourth central moment measures kurtosis.
5. Moments describe shape of distribution fully.
6. They remain consistent under linear transformation.

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10. Probability Distribution (Basic)

Expected value:
$$ E(X)=\sum x_i P(x_i) $$

Variance:
$$ Var(X)=E(X^2)-[E(X)]^2 $$

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Frequently Asked Questions (FAQs)

1. What are these statistics notes mainly designed for?

These statistics notes are mainly designed for students preparing for competitive exams such as JEE Mains, NIMCET, CUET MCA and CET MAH MCA. The notes focus on conceptual clarity, formulas, properties, solved examples and exam-oriented practice.

2. Are these statistics notes useful for NIMCET and other MCA entrance exams?

Yes, these statistics notes are very useful for NIMCET and other MCA entrance exams like CUET MCA and CET MAH MCA, because they cover all important topics such as measures of central tendency, dispersion, correlation, regression, skewness, kurtosis and moments in a clear and structured way.

3. Can JEE Mains aspirants also use these statistics notes?

Yes, JEE Mains aspirants can also use these statistics notes as they are prepared at a level suitable for engineering entrance exams, with sufficient theory, formulas and solved problems for quick revision and practice.

4. Are these notes handwritten in style?

Yes, the content has been created in a simple handwritten-style format so that students feel like they are reading class notes, which makes understanding and memorising statistics concepts easier.

5. Which statistics topics are covered in these notes?

These notes cover major statistics topics such as introduction to statistics, data types, measures of central tendency (mean, median, mode), measures of dispersion (range, mean deviation, variance, standard deviation), coefficient of variation, correlation, regression, skewness, kurtosis and moments.

6. How are moments used in these statistics notes?

Moments are used in these notes to explain the shape of a distribution. Raw moments and central moments are defined, and their role in measuring variance, skewness and kurtosis is explained with formulas and simple examples so that NIMCET and MCA aspirants can understand the concept clearly.

7. What is Aspire Study and how is it related to these notes?

Aspire Study is an online learning platform focused on preparation for exams like NIMCET, JEE Mains, CUET MCA and CET MAH MCA. The statistics notes are part of the study material prepared under Aspire Study to provide high-quality handwritten-style content, practice questions and exam support to students.

8. Do these statistics notes follow the syllabus of NIMCET and CUET MCA?

Yes, the statistics notes are made keeping in mind the latest syllabus of NIMCET, CUET MCA and other MCA entrance exams, so that students can prepare confidently without missing any important topic from the exam point of view.

9. Are there solved examples for each statistics topic?

Yes, for every major statistics topic such as mean, median, mode, variance, standard deviation, correlation, regression, skewness, kurtosis and moments, these notes include step-by-step solved examples to help students understand the application of formulas in problems.

10. Can a beginner in statistics start directly with these notes?

Yes, a beginner can start directly with these notes because the explanation begins from basic definitions and gradually moves towards formulas, properties and examples. The simple handwritten-style presentation makes it easy to follow even for those who are new to statistics.

11. Are these notes sufficient for last-minute revision for NIMCET and MCA exams?

Yes, these statistics notes are concise and well-organised, so they are very helpful for last-minute revision for NIMCET, CUET MCA, CET MAH MCA and similar exams. Students can quickly revise important formulas, properties and key examples before the exam.

12. Does Aspire Study provide other subjects’ notes also for NIMCET and MCA preparation?

Yes, along with statistics notes, Aspire Study provides notes and study material for other important subjects such as mathematics, reasoning, computer awareness and related topics required for NIMCET, CUET MCA, CET MAH MCA and other MCA entrance exams.