Business Analytics (MBA-7302) hptu MBA DETAILED NOTES OF CURRENT SYLLABUS

📘 INTRODUCTION TO BASIC STATISTICS

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🔹 1. MEASURES OF CENTRAL TENDENCY

Yeh wo values hoti hain jo data ko ek single representative value se summarize karti hain.
Common measures: Mean, Median, Mode

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🧮 (A) MEAN (Arithmetic Mean)

(i) Individual Series

Formula:

$$\bar{X} = \frac{\sum X}{N}$$

Example:
X = 10, 20, 30, 40, 50

$$\bar{X} = \frac{10 + 20 + 30 + 40 + 50}{5} = 30$$

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(ii) Discrete Series

Formula:

$$\bar{X} = \frac{\sum fX}{\sum f}$$

Example:

X	f
10	2
20	3
30	5

$$\bar{X} = \frac{(10×2 + 20×3 + 30×5)}{(2+3+5)} = \frac{260}{10} = 26$$

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(iii) Continuous Series

Formula (using class intervals):

$$\bar{X} = A + \frac{\sum f d}{\sum f} \times i$$

Where:
A = Assumed mean,
d = (X - A)/i,
i = Class interval width.

Example:

Class	f	Midpoint (X)	d = (X-A)/i	f*d
0–10	4	5	-2	-8
10–20	6	15	-1	-6
20–30	10	25	0	0
30–40	8	35	1	8
40–50	2	45	2	4

$$\bar{X} = 25 + \frac{(-2)}{30}×10 = 24.33$$

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📏 (B) MEDIAN

Median = Middle value that divides data into two equal parts.

(i) Individual Series

Steps:

1. Arrange data in ascending order.

2. If n is odd → Median = $\frac{n+1}{2}$th item
   If n is even → Median = Average of $\frac{n}{2}$th and $\frac{n}{2}+1$th items.

Example:
X = 10, 15, 20, 25, 30
Median = 3rd item = 20

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(ii) Discrete Series

Steps:

1. Make cumulative frequency (c.f.)

2. Find $\frac{N+1}{2}$th item → see which class or value contains it.

Example:

X	f	c.f
10	2	2
20	3	5
30	5	10

N=10 → $\frac{10+1}{2} = 5.5$th item → lies in value 30
Median = 30

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(iii) Continuous Series

Formula:

$$Median = L + \left(\frac{\frac{N}{2} - C_f}{f}\right) \times i$$

Where:
L = Lower boundary of median class
Cf = cumulative frequency before median class
f = frequency of median class
i = class width

Example:

Class	f	c.f
0–10	4	4
10–20	6	10
20–30	10	20
30–40	8	28

N=28 → N/2 =14 → median class = 20–30

$$Median = 20 + \frac{(14-10)}{10} \times 10 = 24$$

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📊 (C) MODE

Mode = Most frequent or most repeated value.

(i) Individual Series

Value which occurs maximum times.

Example:
10, 20, 20, 25, 30
Mode = 20

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(ii) Discrete Series

Value with highest frequency.

Example:

X	f
10	3
20	7
30	5

Mode = 20

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(iii) Continuous Series

Formula:

$$Mode = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times i$$

Where:
L = Lower boundary of modal class
f₁ = frequency of modal class
f₀ = frequency of class before modal class
f₂ = frequency of class after modal class
i = class width

Example:

Class	f
0–10	5
10–20	8
20–30	12
30–40	10

Modal class = 20–30

$$Mode = 20 + \frac{(12-8)}{2(12)-8-10} \times 10 = 20 + \frac{4}{6} \times 10 = 26.67$$