Statistics

Introduction to Statistics

**Statistics** is the science of averages and their estimates. While measures of central tendency (mean, median, mode) tell us where data points cluster, they do not reveal how the data are **scattered or dispersed** around that central value.

**Example (Batsman Comparison):**

Batsman A scores: 30, 91, 0, 64, 42, 80, 30, 5, 117, 71 → Mean = 53, Median = 53

Batsman B scores: 53, 46, 48, 50, 53, 53, 58, 60, 57, 52 → Mean = 53, Median = 53

Although both have identical mean and median, Batsman A's scores vary from 0 to 117 (highly scattered), while Batsman B's scores cluster between 46 and 60 (tightly grouped). This difference is captured by **measures of dispersion**.

**Key Insight:** Two datasets with the same central tendency can have vastly different distributions. A complete statistical description requires both central tendency AND a measure of how far data points deviate from that center.

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Measures of Dispersion

**Dispersion (or scatter)** is the extent to which observations spread out from a central value. Four main measures of dispersion are:

1. **Range** — simplest, based on max and min values

2. **Quartile Deviation** — (not covered in this chapter)

3. **Mean Deviation** — average absolute deviations from a central value

4. **Standard Deviation** — most rigorous measure (discussed in subsequent sections)

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Range

**Definition:** Range is the difference between the maximum and minimum values in a dataset.

**Formula:** Range = Maximum Value − Minimum Value

**Characteristics:**

Simplest measure of dispersion

Depends only on two extreme values; ignores all middle values

Highly sensitive to outliers

Does not show dispersion about a measure of central tendency

**Example (Batsmen Again):**

Batsman A: Range = 117 − 0 = **117**

Batsman B: Range = 60 − 46 = **14**

Since Batsman A has a much larger range, scores are more scattered. Range alone is insufficient because it doesn't measure how all individual observations deviate from the center.

**Limitation:** Range gives only a rough idea of variability. It tells nothing about how the intermediate values are distributed relative to the central tendency.

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Mean Deviation

Concept and Definition

**Mean Deviation (M.D.)** about a fixed value 'a' is the **average of the absolute values of deviations** of observations from that value.

**Why not use simple deviations?**

When we compute deviations (xi − x̄) from the mean, some are positive and some negative

The sum of all deviations from the mean equals zero: Σ(xi − x̄) = 0

Therefore, the "average" of deviations becomes 0, which is useless for measuring dispersion

**Solution:** Take absolute values |xi − x̄| before averaging

**Formula (General):**

M.D.(a) = [Σ|xi − a|] / n

where:

a = fixed value (usually mean x̄ or median M)

n = number of observations

The absolute value |xi − a| gives the distance of each observation from a

**Two Common Forms:**

1. **M.D. about Mean:** M.D.(x̄) = (1/n) Σ|xi − x̄|

2. **M.D. about Median:** M.D.(M) = (1/n) Σ|xi − M|

**Remark:** Mean deviation can be computed about any measure of central tendency, but mean and median are most practical. Mean deviation about the median is often preferred when median better represents the data.

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Mean Deviation for Ungrouped Data

**Step-by-Step Procedure:**

**Step 1:** Calculate the measure of central tendency (mean or median)

**Step 2:** Find the deviation of each observation from the chosen central tendency:

(xi − a) for all i

**Step 3:** Take absolute values (ignore negative signs):

|x₁ − a|, |x₂ − a|, |x₃ − a|, ..., |xn − a|

**Step 4:** Find the mean of these absolute deviations:

M.D.(a) = [Σ|xi − a|] / n

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Mean Deviation about Mean (Ungrouped Data)

**Example 1:** Find mean deviation about mean for: 6, 7, 10, 12, 13, 4, 8, 12

**Solution:**

**Step 1 - Find Mean:**

x̄ = (6 + 7 + 10 + 12 + 13 + 4 + 8 + 12) / 8 = 72 / 8 = **9**

**Step 2 - Find Deviations:**

| Data | Deviation (xi − x̄) |

|------|-----|

| 6 | 6 − 9 = −3 |

| 7 | 7 − 9 = −2 |

| 10 | 10 − 9 = +1 |

| 12 | 12 − 9 = +3 |

| 13 | 13 − 9 = +4 |

| 4 | 4 − 9 = −5 |

| 8 | 8 − 9 = −1 |

| 12 | 12 − 9 = +3 |

**Step 3 - Absolute Values:**

|−3|, |−2|, |+1|, |+3|, |+4|, |−5|, |−1|, |+3| = 3, 2, 1, 3, 4, 5, 1, 3

**Step 4 - Mean of Absolute Deviations:**

M.D.(x̄) = (3 + 2 + 1 + 3 + 4 + 5 + 1 + 3) / 8 = 22 / 8 = **2.75**

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Mean Deviation about Median (Ungrouped Data)

**Example 2:** Find mean deviation about median for: 3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21

**Solution:**

**Step 1 - Arrange in Order:**

3, 3, 4, 5, 7, 9, 10, 12, 18, 19, 21 (n = 11, which is odd)

**Step 2 - Find Median:**

Median position = [(n + 1)/2]th = [(11 + 1)/2]th = 6th observation = **9**

**Step 3 - Absolute Deviations from Median:**

| Data | |xi − M| |

|------|-----|

| 3 | \|3 − 9\| = 6 |

| 3 | 6 |

| 4 | 5 |

| 5 | 4 |

| 7 | 2 |

| 9 | 0 |

| 10 | 1 |

| 12 | 3 |

| 18 | 9 |

| 19 | 10 |

| 21 | 12 |

Sum = 6 + 6 + 5 + 4 + 2 + 0 + 1 + 3 + 9 + 10 + 12 = **58**

**Step 4 - Calculate M.D.:**

M.D.(M) = 58 / 11 = **5.27** (approximately)

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Mean Deviation for Grouped Data

When data are presented as frequency distributions, we use the same formulas but weight deviations by frequencies.

#### For Discrete Frequency Distribution

| xi | x₁ | x₂ | ... | xn |

|----|-----|-----|-----|-----|

| fi | f₁ | f₂ | ... | fn |

**Mean:** x̄ = (Σ fi·xi) / N, where N = Σ fi

**M.D. about Mean:**

M.D.(x̄) = [Σ fi|xi − x̄|] / N

**M.D. about Median:**

M.D.(M) = [Σ fi|xi − M|] / N

**Finding Median for Discrete Distribution:**

Arrange data in ascending order

Calculate cumulative frequencies

The observation whose cumulative frequency is **equal to or just greater than N/2** is the median

**Example 3:** Find mean deviation about mean:

| xi | 2 | 5 | 6 | 8 | 10 | 12 |

|----|---|---|---|----|-----|-----|

| fi | 2 | 8 | 10| 7 | 8 | 5 |

**Solution:**

| xi | fi | fi·xi | \|xi − x̄\| | fi\|xi − x̄\| |

|----|-----|-------|---------|----------|

| 2 | 2 | 4 | 5.5 | 11 |

| 5 | 8 | 40 | 2.5 | 20 |

| 6 | 10 | 60 | 1.5 | 15 |

| 8 | 7 | 56 | 0.5 | 3.5 |

| 10 | 8 | 80 | 2.5 | 20 |

| 12 | 5 | 60 | 4.5 | 22.5 |

| **Total** | **40** | **300** | | **92** |

Mean: x̄ = 300 / 40 = **7.5**

M.D.(x̄) = 92 / 40 = **2.3**

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Mean Deviation for Continuous Frequency Distribution

For class intervals, use the **midpoint (xi)** of each class interval as the representative value.

**Formula:**

M.D.(x̄) = [Σ fi|xi − x̄|] / N

**Example 4:** Find mean deviation about mean:

| Marks | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 | 60-70 | 70-80 |

|-------|-------|-------|-------|-------|-------|-------|-------|

| Students | 2 | 3 | 8 | 14 | 8 | 3 | 2 |

**Solution:**

| Class | fi | Midpoint (xi) | fi·xi | \|xi − x̄\| | fi\|xi − x̄\| |

|-------|-----|-------|--------|---------|----------|

| 10-20 | 2 | 15 | 30 | 30 | 60 |

| 20-30 | 3 | 25 | 75 | 20 | 60 |

| 30-40 | 8 | 35 | 280 | 10 | 80 |

| 40-50 | 14 | 45 | 630 | 0 | 0 |

| 50-60 | 8 | 55 | 440 | 10 | 80 |

| 60-70 | 3 | 65 | 195 | 20 | 60 |

| 70-80 | 2 | 75 | 150 | 30 | 60 |

| **Total** | **40** | | **1800** | | **400** |

Mean: x̄ = 1800 / 40 = **45**

M.D.(x̄) = 400 / 40 = **10**

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Step-Deviation Method for Computing Mean

To simplify calculations, use an **assumed mean (a)** close to the actual mean and a **common factor (h)** dividing the deviations.

**Step-Deviation:** di = (xi − a) / h

**Formula:** x̄ = a + h × [Σ fi·di / N]

This reduces arithmetic but doesn't change the M.D. calculation itself—only the mean computation is simplified.

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Mean Deviation about Median (Continuous Distribution)

**Process:**

1. Find the median class (cumulative frequency ≥ N/2)

2. Apply median formula: Median = l + [(N/2 − C) / f] × h

3. Calculate |xi − Median| for each class midpoint

4. Compute M.D.(M) = [Σ fi|xi − M|] / N

**Example 5:**

| Class | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

|-------|------|-------|-------|-------|-------|-------|

| Frequency | 6 | 7 | 15 | 16 | 4 | 2 |

**Solution:**

| Class | fi | c.f. | xi | \|xi − Median\| | fi\|xi − Median\| |

|-------|-----|------|-----|---------|------------|

| 0-10 | 6 | 6 | 5 | 23 | 138 |

| 10-20 | 7 | 13 | 15 | 13 | 91 |

| 20-30 | 15 | 28 | 25 | 3 | 45 |

| 30-40 | 16 | 44 | 35 | 7 | 112 |

| 40-50 | 4 | 48 | 45 | 17 | 68 |

| 50-60 | 2 | 50 | 55 | 27 | 54 |

| | **50** | | | | **508** |

Median class = 20-30 (since c.f. = 28 > N/2 = 25)

Median = 20 + [(25 − 13) / 15] × 10 = 20 + 8 = **28**

M.D.(M) = 508 / 50 = **10.16**

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Limitations of Mean Deviation

1. **When median is not representative:** In highly skewed data, the median is not a good central tendency, so M.D. about median is unreliable

2. **Mathematically problematic:** The use of absolute values makes mean deviation difficult to manipulate algebraically compared to squared deviations

3. **Sum of deviations from median is always less than from mean:** Σ|xi − M| < Σ|xi − x̄| (mathematically proven). This means M.D. about mean is not ideal from a scientific standpoint

4. **Less useful in advanced statistics:** Mean deviation cannot be easily used in further statistical analysis, hypothesis testing, or inferential statistics

**Conclusion:** While mean deviation provides a straightforward measure of dispersion, **standard deviation** (which uses squared deviations) is preferred in professional statistics because it overcomes these limitations and has superior algebraic properties.

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Key Exam Points to Remember

**Mean deviation must use absolute values** to prevent positive and negative deviations from canceling

**Always calculate the measure of central tendency first** before finding deviations

**For discrete/continuous grouped data:** weight each deviation by its frequency

**Use midpoints** for class intervals in continuous distributions

**The step-deviation method only simplifies the mean calculation**, not the M.D. itself

**Common exam errors:** forgetting absolute values, using wrong median, arithmetic mistakes in totaling

These notes cover the **complete Chapter 13 on Statistics** for CBSE Class 11, providing all definitions, formulas, step-by-step procedures, worked examples, and exam-critical insights needed for board preparation.