Measurement of Scale

Definition

Measurement of scale refers to the system or framework used to categorize, quantify, and classify data based on the properties and relationships of the values assigned to variables. It defines the nature of information contained in data and determines which statistical operations and analyses are appropriate.

In other words, a scale of measurement is a classification that describes the nature of information within the numbers assigned to variables.

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Types of Measurement Scales

There are four main types of measurement scales:

1. Nominal Scale

• Definition: The most basic level of measurement where data is categorized into distinct, non-overlapping categories with no inherent order or ranking.
• Characteristics:
  • Categories are mutually exclusive
  • No numerical relationship between categories
  • No ordering or hierarchy
• Examples: Gender (Male/Female), Religion, Blood type (A, B, AB, O), Color
• Allowed Operations: Counting, frequency, mode

2. Ordinal Scale

• Definition: Data is organized into categories that have a natural order or ranking, but the intervals between categories are not equal or meaningful.
• Characteristics:
  • Categories can be ranked or ordered
  • Differences between ranks are not uniform
  • Relative position matters, not absolute value
• Examples: Educational level (Primary, Secondary, Bachelor's, Master's), Customer satisfaction (Poor, Fair, Good, Excellent), Military rank
• Allowed Operations: Median, mode, ranking, correlation

3. Interval Scale

• Definition: Numerical scale where the distance between any two adjacent units is equal, but there is no true zero point (zero is arbitrary).
• Characteristics:
  • Equal intervals between values
  • No true/absolute zero
  • Allows meaningful differences
• Examples: Temperature (Celsius/Fahrenheit), Calendar years, IQ scores
• Allowed Operations: Mean, standard deviation, addition, subtraction

4. Ratio Scale

• Definition: The highest level of measurement with equal intervals between values AND a true/absolute zero point, allowing for meaningful ratios.
• Characteristics:
  • Equal intervals between values
  • True zero point exists
  • Ratios are meaningful
  • Most flexible scale
• Examples: Height, Weight, Age, Income, Distance, Time
• Allowed Operations: All statistical operations including mean, median, mode, standard deviation, multiplication, division

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Summary Table

Correlation Coefficient: Definition and Calculation

Definition

The correlation coefficient is a statistical measure that expresses the degree and direction of the linear relationship between two variables. It ranges from $-1$ to $+1$:

• $+1$ = perfect positive correlation
• $-1$ = perfect negative correlation
• $0$ = no linear correlation

Pearson's correlation coefficient:

$$r = \frac{n\sum XY - \sum X \sum Y}{\sqrt{[n\sum X^2 - (\sum X)^2][n\sum Y^2 - (\sum Y)^2]}}$$

Given Data (n = 10)

Summations (verified)

• $\sum X = 23+27+28+28+29+30+31+33+35+36 = 300$
• $\sum Y = 18+20+27+21+29+27+27+29+28+29 = 255$
• $\sum XY = 414+540+756+588+841+810+837+957+980+1044 = 7767$
• $\sum X^2 = 529+729+784+784+841+900+961+1089+1225+1296 = 9138$
• $\sum Y^2 = 324+400+729+441+841+729+729+841+784+841 = 6659$

Note: the correct sums are $\sum X = 300$ (not 310), $\sum X^2 = 9138$ (not 9738) and $\sum Y^2 = 6659$ (not 6859). Rechecking $\sum Y^2$: $324+400=724$, $+729=1453$, $+441=1894$, $+841=2735$, $+729=3464$, $+729=4193$, $+841=5034$, $+784=5818$, $+841=6659$. Confirmed $6659$.

Substitution

Numerator: $$n\sum XY - \sum X\sum Y = 10(7767) - (300)(255) = 77670 - 76500 = 1170$$

X-term: $$n\sum X^2 - (\sum X)^2 = 10(9138) - (300)^2 = 91380 - 90000 = 1380$$

Y-term: $$n\sum Y^2 - (\sum Y)^2 = 10(6659) - (255)^2 = 66590 - 65025 = 1565$$

$$r = \frac{1170}{\sqrt{1380 \times 1565}} = \frac{1170}{\sqrt{2159700}} = \frac{1170}{1469.59}$$

$$\boxed{r \approx 0.796}$$

Interpretation

$r \approx +0.796$ indicates a strong positive correlation between husbands' and wives' ages. As the husband's age increases, the wife's age also tends to increase. This is consistent with real-world marriage patterns. The coefficient of determination $r^2 \approx 0.634$, so about 63% of the variation in wife's age is explained by husband's age.

A result of $r \approx -0.646$ comes from using $\sum X = 310$ instead of $300$, together with wrong squared sums, which flips the sign of the numerator.

Model Answer: Binomial Probability

STEP 1 - Given Data

• Probability of thick enough coating: $p = 0.70$
• Probability of not thick enough: $q = 1 - p = 0.30$
• Sample size: $n = 15$

Binomial distribution: $P(X = r) = \binom{n}{r} p^r q^{n-r}$

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STEP 2 - Solve

(1) P(at least 12) = P(X ≥ 12)

$$P(X \geq 12) = P(12) + P(13) + P(14) + P(15)$$

P(X = 12): $$\binom{15}{12}(0.7)^{12}(0.3)^{3} = 455 \times 0.0138413 \times 0.027$$

$(0.7)^{12} = 0.01384128$

$= 455 \times 0.01384128 \times 0.027 = 0.17004$

P(X = 13): $$\binom{15}{13}(0.7)^{13}(0.3)^{2} = 105 \times 0.00968889 \times 0.09$$

$(0.7)^{13} = 0.00968889$

$= 105 \times 0.00968889 \times 0.09 = 0.09156$

P(X = 14): $$\binom{15}{14}(0.7)^{14}(0.3)^{1} = 15 \times 0.00678223 \times 0.3$$

$(0.7)^{14} = 0.00678223$

$= 15 \times 0.00678223 \times 0.3 = 0.03052$

P(X = 15): $$\binom{15}{15}(0.7)^{15}(0.3)^{0} = 1 \times 0.00474756 \times 1 = 0.00475$$

Total: $$P(X \geq 12) = 0.17004 + 0.09156 + 0.03052 + 0.00475 = 0.2969$$

$$\boxed{P(X \geq 12) \approx 0.2969}$$

(2) P(exactly 10) = P(X = 10)

$$P(X = 10) = \binom{15}{10}(0.7)^{10}(0.3)^{5}$$

$\binom{15}{10} = 3003$

$(0.7)^{10} = 0.0282475$

$(0.3)^{5} = 0.00243$

$$= 3003 \times 0.0282475 \times 0.00243$$

$$= 3003 \times 0.00006864 = 0.2061$$

$$\boxed{P(X = 10) \approx 0.2061}$$

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Summary of Answers

1. P(at least 12) ≈ 0.2969 or 29.69%
2. P(exactly 10) ≈ 0.2061 or 20.61%

Model Answer: Fourth Central Moment and Kurtosis

Step 1 - Given Data

• Standard deviation: $\sigma = 7$
• Distribution is symmetric
• Required: value of the fourth central moment $\mu_4$ for the distribution to be (i) mesokurtic, (ii) platykurtic, (iii) leptokurtic

Step 2 - Solve

Key Relationship

The coefficient of kurtosis is: $$\beta_2 = \frac{\mu_4}{\mu_2^2} = \frac{\mu_4}{\sigma^4}$$

since $\mu_2 = \sigma^2$.

Rearranging: $$\mu_4 = \beta_2 \times \sigma^4$$

Compute $\sigma^4$

$$\sigma^4 = 7^4 = 2401$$

(i) Mesokurtic

Same peakedness as normal distribution: $\beta_2 = 3$. $$\mu_4 = 3 \times 2401 = \boxed{7203}$$

(ii) Platykurtic

Flatter than normal: $\beta_2 < 3$. $$\mu_4 < 3 \times 2401 = \boxed{7203} \quad (\mu_4 < 7203)$$

(iii) Leptokurtic

More peaked than normal: $\beta_2 > 3$. $$\mu_4 > 3 \times 2401 = \boxed{7203} \quad (\mu_4 > 7203)$$

Summary Table

The symmetry condition confirms odd central moments are zero, but it does not affect the kurtosis computation, which depends only on $\mu_4$ and $\sigma$.