Arithmetic Mean – Calculation & Properties ➗📊

Arithmetic Mean (A.M.) is the most widely used and most important average in statistics. It is simple, logical, and mathematically powerful.

---

Definition

Arithmetic Mean is obtained by dividing the sum of all observations by the number of observations.

Formula

Mean = (ΣX) / N

Where:

• X = individual values
• N = number of observations

---

Methods of Calculating Mean 🧮

1. Direct Method

X̄ = ΣX / N

Example: Values: 10, 20, 30, 40 ΣX = 100 N = 4

Mean = 100 / 4 = 25

---

2. Shortcut Method (Assumed Mean Method)

Used when values are large or inconvenient.

Steps:

1. Choose an assumed mean A.
2. Compute deviations: d = X – A.
3. Apply:

X̄ = A + (Σd / N)

Example: Values = 45, 50, 55 A = 50 d = -5, 0, +5

Mean = 50 + (0 / 3) = 50

---

3. Step-Deviation Method

Used for grouped data with equal class intervals.

X̄ = A + h * (Σu / N)

Where u = (X – A) / h and h = class width.

---

Mean for Grouped Data

Mean = (ΣfX) / (Σf)

Example Table

Class	Midpoint X	f	fX
10–20	15	4	60
20–30	25	6	150
30–40	35	10	350

ΣfX = 560
Σf = 20
Mean = 560 / 20 = 28

---

Properties of Arithmetic Mean ⭐

1. Rigidly defined — always unique.
2. Uses all observations.
3. Algebraic property — used in SD, correlation, regression.
4. Affected by extreme values.
5. Balancing property:

Σ (X – X̄) = 0

---

When Mean Is Appropriate ✔️

• Data is continuous.
• No extreme outliers.
• Mathematical operations required.

When Mean Is Not Suitable ❌

• Highly skewed data.
• Open-ended classes.
• Qualitative data.

---

Solved Example (Grouped Data)

Find the mean:

Class	Midpoint X	f
0–10	5	3
10–20	15	7
20–30	25	10

ΣfX = 3×5 + 7×15 + 10×25
ΣfX = 15 + 105 + 250 = 370
Σf = 20
Mean = 370 / 20 = 18.5

---

Summary ✨

• Arithmetic Mean = ΣX / N
• Methods: direct, shortcut, step-deviation, grouped data.
• Properties: rigid, algebraic, uses all values, affected by extremes.
• Widely used in business & economics.

---

Quiz Time 🎯