Study guide
5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions
PRC Registered Electrical Engineer REE
Track your progress as you read
Finished sections get a check mark.
1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions
Key Takeaways Explain This Topic AI
Engineering Data Analysis is the single largest Mathematics topic in the PRC Enhanced TOS at 5.00% of the exam and 20 of the 100 Mathematics items.
Sample variance divides by (n − 1), not n; using the population divisor on sample data is the most common numerical slip in REE data-analysis items.
The binomial distribution applies to a fixed number of independent trials with constant success probability: mean = np, variance = npq.
The Poisson distribution models events per unit time or length with mean = variance = λ, making it the standard model for line faults per 100 km-year.
For a normal distribution, z = (x − μ)/σ, and roughly 68%, 95% and 99.7% of values lie within 1, 2 and 3 standard deviations of the mean.
Last updated: August 2026
5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions
Engineering Data Analysis is topic G of the Mathematics subject in the PRC Enhanced Table of Specifications (PRBEE Resolution No. 40, s. 2024). At 5.00% of the total examination and 20 of the 100 Mathematics items , it is the largest single Mathematics topic — larger than Calculus 1, Calculus 2 or Differential Equations individually. It entered the syllabus through CHED Memorandum Order No. 88, s. 2017, which made Engineering Data Analysis a required BSEE course.
1. Measures of Central Tendency
- Measure Definition When it is the right answer
- Arithmetic mean x ˉ \bar{x} x ˉ 1 n ∑ x i \dfrac{1}{n}\sum x_i n 1 ∑ x i Symmetric data; the default
- Median Middle value of the ordered set Skewed data or outliers present
- Mode Most frequent value Categorical data; most-common conductor size
Weighted mean ∑ w i x i ∑ w i \dfrac{\sum w_i x_i}{\sum w_i} ∑ w i ∑ w i x i REE general weighted average of subject grades
Geometric mean x 1 x 2 ⋯ x n n \sqrt[n]{x_1 x_2 \cdots x_n} n x 1 x 2 ⋯ x n Average growth or load-growth rates
Root-mean-square 1 n ∑ x i 2 \sqrt{\dfrac{1}{n}\sum x_i^2} n 1 ∑ x i 2 Effective value of a current or voltage waveform
The RMS entry is not incidental: the same statistic that engineers call "effective value" is a descriptive statistic, and the REE examination asks for it in both languages.
2. Measures of Dispersion
Range = x max − x min = x_{\max} - x_{\min} = x m a x − x m i n — quick but discards all interior information.
Variance and standard deviation — note the divisor carefully: σ 2 = ∑ ( x i − μ ) 2 N (population) s 2 = ∑ ( x i − x ˉ ) 2 n − 1 (sample) \sigma^2 = \frac{\sum (x_i - \mu)^2}{N} \quad \text{(population)} \qquad\qquad s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} \quad \text{(sample)} σ 2 = N ∑ ( x i − μ ) 2 (population) s 2 = n − 1 ∑ ( x i − x ˉ ) 2 (sample)
The ( n − 1 ) (n-1) ( n − 1 ) divisor is Bessel's correction ; it makes s 2 s^2 s 2 an unbiased estimator of σ 2 \sigma^2 σ 2 . Applying the population formula to sample data is the most frequent arithmetic error in this topic, and the examiners supply both values as distractors.
Coefficient of variation expresses dispersion as a dimensionless ratio, letting you compare the spread of a 220 V measurement set against a 69 kV set: C V = s x ˉ × 100 % CV = \frac{s}{\bar{x}} \times 100\% C V = x ˉ s × 100%
- 3. Discrete Probability Distributions
- Binomial Distribution
Fixed n n n independent trials, constant success probability p p p , q = 1 − p q = 1-p q = 1 − p : P ( X = k ) = ( n k ) p k q n − k , μ = n p , σ 2 = n p q P(X = k) = \binom{n}{k} p^{k} q^{\,n-k}, \qquad \mu = np, \qquad \sigma^2 = npq P ( X = k ) = ( k n ) p k q n − k , μ = n p , σ 2 = n pq
Use it for: the number of defective breakers in a batch of 20, the number of successful reclosures out of 5 attempts.
Poisson Distribution
Events occurring independently at a constant average rate λ \lambda λ over an interval of time, length or area: P ( X = k ) = λ k e − λ k ! , μ = σ 2 = λ P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}, \qquad \mu = \sigma^2 = \lambda P ( X = k ) = k ! λ k e − λ , μ = σ 2 = λ
The defining signature — mean equals variance — is itself an examinable fact. Use it for: faults per 100 circuit-km per year, customer interruptions per feeder per month, lightning strikes per tower per year.
Poisson as a Limit of the Binomial
When n n n is large and p p p small with n p = λ np = \lambda n p = λ moderate, the binomial converges to the Poisson. This is why rare-event reliability problems are Poisson problems.
- 4. Continuous Probability Distributions
- Exponential Distribution
Models the time between Poisson events, i.e. time to failure at constant hazard rate λ \lambda λ : f ( t ) = λ e − λ t , R ( t ) = P ( T > t ) = e − λ t , MTBF = 1 λ f(t) = \lambda e^{-\lambda t}, \qquad R(t) = P(T > t) = e^{-\lambda t}, \qquad \text{MTBF} = \frac{1}{\lambda} f ( t ) = λ e − λ t , R ( t ) = P ( T > t ) = e − λ t , MTBF = λ 1
The exponential distribution is memoryless : a component that has survived 10 000 hours has the same remaining-life distribution as a new one. This is the assumption behind constant-failure-rate reliability blocks.
Normal (Gaussian) Distribution z = x − μ σ z = \frac{x - \mu}{\sigma} z = σ x − μ
- Interval Approximate probability
- μ ± 1 σ \mu \pm 1\sigma μ ± 1 σ 68.27 % 68.27\% 68.27%
- μ ± 2 σ \mu \pm 2\sigma μ ± 2 σ 95.45 % 95.45\% 95.45%
- μ ± 3 σ \mu \pm 3\sigma μ ± 3 σ 99.73 % 99.73\% 99.73%
- μ ± 1.645 σ \mu \pm 1.645\sigma μ ± 1.645 σ 90 % 90\% 90% (two-tailed)
- μ ± 1.96 σ \mu \pm 1.96\sigma μ ± 1.96 σ 95 % 95\% 95% (two-tailed)
The z = 1.96 z = 1.96 z = 1.96 and z = 1.645 z = 1.645 z = 1.645 values recur in every acceptance-limit and confidence-interval item, so commit them to memory.
- Solved Board Exam Examples
- Example 1: Sample Standard Deviation of Insulation Test Readings
Five megger readings on a motor winding give 480, 495, 510, 470 and 505 MΩ. Compute the sample standard deviation.
Solution. The mean is x ˉ = 480 + 495 + 510 + 470 + 505 5 = 2460 5 = 492 M Ω \bar{x} = \frac{480 + 495 + 510 + 470 + 505}{5} = \frac{2460}{5} = 492 \text{ M}\Omega x ˉ = 5 480 + 495 + 510 + 470 + 505 = 5 2460 = 492 M Ω
Deviations and their squares: ( − 12 ) 2 = 144 (-12)^2 = 144 ( − 12 ) 2 = 144 , ( 3 ) 2 = 9 (3)^2 = 9 ( 3 ) 2 = 9 , ( 18 ) 2 = 324 (18)^2 = 324 ( 18 ) 2 = 324 , ( − 22 ) 2 = 484 (-22)^2 = 484 ( − 22 ) 2 = 484 , ( 13 ) 2 = 169 (13)^2 = 169 ( 13 ) 2 = 169 ; the sum is 1130 1130 1130 . s = 1130 5 − 1 = 282.5 = 16.81 M Ω s = \sqrt{\frac{1130}{5 - 1}} = \sqrt{282.5} = \boxed{16.81 \text{ M}\Omega} s = 5 − 1 1130 = 282.5 = 16.81 M Ω
Dividing by 5 instead of 4 would give 15.03 MΩ — the classic distractor.
Example 2: Poisson Fault Rate on a Distribution Feeder
A 13.8 kV feeder experiences an average of 3 faults per year. What is the probability of exactly 2 faults in a given year, and of no faults at all?
Solution. With λ = 3 \lambda = 3 λ = 3 : P ( X = 2 ) = 3 2 e − 3 2 ! = 9 ( 0.049787 ) 2 = 0.224 P(X = 2) = \frac{3^{2} e^{-3}}{2!} = \frac{9(0.049787)}{2} = \boxed{0.224} P ( X = 2 ) = 2 ! 3 2 e − 3 = 2 9 ( 0.049787 ) = 0.224 P ( X = 0 ) = 3 0 e − 3 0 ! = e − 3 = 0.0498 P(X = 0) = \frac{3^{0} e^{-3}}{0!} = e^{-3} = \boxed{0.0498} P ( X = 0 ) = 0 ! 3 0 e − 3 = e − 3 = 0.0498
A fault-free year on this feeder occurs only about once per twenty years.
Example 3: Normal Acceptance Limit on Transformer Losses
Distribution transformer no-load losses are normally distributed with μ = 150 \mu = 150 μ = 150 W and σ = 12 \sigma = 12 σ = 12 W. A purchase specification rejects any unit exceeding 174 W. What fraction is rejected?
Solution. z = 174 − 150 12 = 2.00 z = \frac{174 - 150}{12} = 2.00 z = 12 174 − 150 = 2.00
From the standard normal table, P ( Z ≤ 2.00 ) = 0.9772 P(Z \le 2.00) = 0.9772 P ( Z ≤ 2.00 ) = 0.9772 , so P ( X > 174 ) = 1 − 0.9772 = 0.0228 or 2.28 % P(X > 174) = 1 - 0.9772 = \boxed{0.0228 \text{ or } 2.28\%} P ( X > 174 ) = 1 − 0.9772 = 0.0228 or 2.28%
About 23 units in every 1 000 fail the loss guarantee.