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4.3 Descriptive Statistics and Data Interpretation

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Adamson University College Entrance Test AdUCET

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1 Introduction & Exam Overview

1.1 AdUCET Overview, Format, and Admission Process 1.2 Test-Taking and Time-Management Strategies

2 Chapter 2: Mathematics I — Numbers, Algebra, and Functions

2.1 Number Sense, Operations, and Fraction–Decimal–Percent Conversions 2.2 Algebra, Polynomials, and Linear/Quadratic Equations 2.3 Inequalities, Absolute Value, and Systems of Equations 2.4 Radicals, Rational Expressions, and Scientific Notation 2.5 Coordinate Geometry, Lines, and Functions

3 Chapter 3: Mathematics II — Geometry, Measurement, and Trigonometry

3.1 Plane Geometry: Angles, Triangles, Polygons, and Circles 3.2 Solid Mensuration, Measurement, and Unit Conversion 3.3 Right-Triangle Trigonometry and Applications

4 Chapter 4: Mathematics III — Sequences, Proportion, and Data

4.1 Arithmetic and Geometric Sequences and Series 4.2 Ratio, Proportion, Percentage, Work, and Mixture Problems 4.3 Descriptive Statistics and Data Interpretation 4.4 Probability: Rules, Compound Events, and Conditional Probability 4.5 Counting Principles: Permutations and Combinations

5 Chapter 5: English I — Reading Comprehension and Vocabulary

5.1 Reading Comprehension: Main Idea, Purpose, and Passage Structure 5.2 Inference, Supporting Detail, and Author's Tone 5.3 Word Roots, Prefixes, and Suffixes 5.4 Vocabulary in Context, Synonyms, Antonyms, and Confusables

6 Chapter 6: English II — Grammar, Usage, and Written Expression

6.1 Subject-Verb Agreement and Verb Tenses 6.2 Pronouns, Modifiers, and Parallel Structure 6.3 Sentence Boundaries, Clauses, and Punctuation 6.4 Sentence Correction, Redundancy, and Idiomatic Usage 6.5 Verbal Analogies and Relationship Bridges 6.6 Paragraph Organization, Coherence, and Transitions

7 Chapter 7: General Science I — Biology and Chemistry

7.1 Cell Biology, Transport, and Cellular Energetics 7.2 Genetics, Heredity, and Molecular Biology 7.3 Human Body Systems, Ecology, and Evolution 7.4 Matter, Phase Changes, and Atomic Structure 7.5 Periodic Trends, Chemical Bonding, Reactions, Acids and Bases

8 Chapter 8: General Science II — Physics and Earth Science

8.1 Mechanics: Motion, Newton's Laws, Work, and Energy 8.2 Waves, Sound, Light, Heat, Electricity, and Magnetism 8.3 Earth's Structure, Plate Tectonics, and the Rock Cycle 8.4 The Atmosphere, Weather, the Water Cycle, and Astronomy

9 Chapter 9: Aptitude and Abstract Reasoning

9.1 Figure Series and Pattern Progression 9.2 Matrix Reasoning and Figural Analogies 9.3 Spatial Reasoning: Rotation, Reflection, and Chirality 9.4 Paper Folding, Cube Nets, and Orthographic Views 9.5 Deductive Reasoning, Syllogisms, and Venn Diagrams 9.6 Number, Letter, and Alphanumeric Series 9.7 Odd-One-Out, Classification, and Logic Puzzles

4.3 Descriptive Statistics and Data Interpretation

Key Takeaways Explain This Topic AI

The grouped-data mean is the sum of (frequency x class midpoint) divided by the total frequency, where each midpoint is (lower limit + upper limit) / 2 — never average the class limits of the whole table.

Outliers move the mean but not the median: five allowances of P100, P120, P150, P150 and P180 average P140, but adding one classmate at P900 raises the mean to about P266.67 while the median stays at P150.

Population variance divides the sum of squared deviations by N; sample variance divides by n - 1, which always makes the sample value the larger of the two.

Adding the same constant to every value shifts the mean by that constant and leaves the standard deviation and variance unchanged; multiplying every value by k multiplies the mean and standard deviation by k and the variance by k squared.

Under the empirical rule a roughly normal set with mean 78 and standard deviation 6 puts about 68% of scores between 72 and 84, about 95% between 66 and 90, and about 16% above 84.

Last updated: August 2026

4.3 Descriptive Statistics and Data Interpretation

Candidate reports describe the AdUCET mathematics items as leaning unusually hard on data handling: a frequency table to average, a graph to read, a median to locate, a claim about spread to judge. These are among the fastest marks available because the arithmetic stays small — but only if you can tell instantly which summary the item is asking for. This section builds that reflex, from the three kinds of mean through the five-number summary to the shape of a distribution.

  • 1. Measures of Centre
  • 1.1 The mean and its three faces

The arithmetic mean is the total divided by the count: x ˉ = ∑ x i n \bar{x} = \frac{\sum x_i}{n} x ˉ = n ∑ x i ​ ​

Two variants dominate entrance-test items.

Weighted mean — used when values carry unequal importance, exactly as in a DepEd class record where written work, performance tasks and the quarterly assessment are worth different percentages: x ˉ w = ∑ w i x i ∑ w i \bar{x}_w = \frac{\sum w_i x_i}{\sum w_i} x ˉ w ​ = ∑ w i ​ ∑ w i ​ x i ​ ​

Grouped mean — used when the raw scores are gone and only a frequency table survives. Each class midpoint stands in for every score in that class: x ˉ = ∑ f i m i ∑ f i , m i = lower limit + upper limit 2 \bar{x} = \frac{\sum f_i m_i}{\sum f_i}, \qquad m_i = \frac{\text{lower limit} + \text{upper limit}}{2} x ˉ = ∑ f i ​ ∑ f i ​ m i ​ ​ , m i ​ = 2 lower limit + upper limit ​

1.2 Median

The median is the middle value of ordered data, so sort first — always. With n n n odd the median is the value in position ( n + 1 ) / 2 (n+1)/2 ( n + 1 ) /2 . With n n n even it is the mean of the values in positions n / 2 n/2 n /2 and n / 2 + 1 n/2 + 1 n /2 + 1 . For six monthly water bills of ₱285, ₱295, ₱320, ₱350, ₱380 and ₱410, the median is ( 320 + 350 ) / 2 = PHP 335 (320 + 350)/2 = \text{PHP }335 ( 320 + 350 ) /2 = PHP 335 — a figure that is not itself in the list, which is perfectly normal.

1.3 Mode and modality

The mode is the value that occurs most often. A data set is unimodal (one clear mode), bimodal (two values tie for the top frequency), multimodal , or has no mode when every value appears equally often. The mode is the only measure of centre that works on categorical data — favourite jeepney route, blood type, senior high strand.

1.4 Choosing the right summary

  • Situation Best measure Why
  • Roughly symmetric numeric data Mean Uses every observation
  • Incomes, land prices, one huge value Median Resistant to extremes
  • Categorical data (strand, colour, brand) Mode Mean and median are undefined
  • Components with different percentages Weighted mean Respects the weights
  • Data already collapsed into class intervals Grouped mean Raw values are unavailable
  • 1.5 Outliers move the mean, not the median

Five students report daily allowances of ₱100, ₱120, ₱150, ₱150 and ₱180. The total is ₱700, so the mean is ₱140, the median is ₱150 and the mode is ₱150. Add one classmate whose allowance is ₱900: the total becomes ₱1,600 and the mean leaps to 1600 / 6 ≈ PHP 266.67 1600/6 \approx \text{PHP }266.67 1600/6 ≈ PHP 266.67 , a figure larger than five of the six actual allowances. The median only moves from ₱150 to ( 150 + 150 ) / 2 = PHP 150 (150 + 150)/2 = \text{PHP }150 ( 150 + 150 ) /2 = PHP 150 — it does not move at all. This single fact is why newspapers quote median salaries and median household income.

2. Worked Example 1 — a frequency table from end to end

Problem. Forty Grade 12 students sat a 40-item mathematics review test. Find the mean, the modal class, the median class, and the central angle that the class 26–30 would occupy in a pie chart.

  • Class interval Frequency f f f Midpoint m m m f × m f \times m f × m Cumulative f f f
  • 11–15 1 13 13 1
  • 16–20 6 18 108 7
  • 21–25 9 23 207 16
  • 26–30 12 28 336 28
  • 31–35 8 33 264 36
  • 36–40 4 38 152 40
  • Total 40 — 1,080 —

Midpoints. m 1 = ( 11 + 15 ) / 2 = 13 m_1 = (11 + 15)/2 = 13 m 1 ​ = ( 11 + 15 ) /2 = 13 ; every later midpoint adds the class width of 5.

Mean. x ˉ = 1080 / 40 = 27.0 \bar{x} = 1080 / 40 = 27.0 x ˉ = 1080/40 = 27.0 marks. Note that 27 is not the average of the midpoints (25.5); the frequencies pull it upward.

Modal class. The largest frequency is 12, so the modal class is 26–30 .

Median class. With n = 40 n = 40 n = 40 the median sits between the 20th and 21st ordered scores. The cumulative frequency reaches only 16 by the end of 21–25 but 28 by the end of 26–30, so both scores lie in 26–30 .

Pie-chart angle. Relative frequency of 26–30 is 12 / 40 = 0.30 12/40 = 0.30 12/40 = 0.30 , so the central angle is 0.30 × 360 ∘ = 108 ∘ 0.30 \times 360^\circ = 108^\circ 0.30 × 36 0 ∘ = 10 8 ∘ .

3. Measures of Spread

Range = x max ⁡ − x min ⁡ = x_{\max} - x_{\min} = x m a x ​ − x m i n ​ . Simple, but a single extreme value destroys it.

Quartiles. Q 1 Q_1 Q 1 ​ is the median of the lower half, Q 2 Q_2 Q 2 ​ is the median itself, Q 3 Q_3 Q 3 ​ is the median of the upper half. The interquartile range is IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 ​ − Q 1 ​ — the width of the middle 50% of the data. A value below Q 1 − 1.5 ( IQR ) Q_1 - 1.5(\text{IQR}) Q 1 ​ − 1.5 ( IQR ) or above Q 3 + 1.5 ( IQR ) Q_3 + 1.5(\text{IQR}) Q 3 ​ + 1.5 ( IQR ) is conventionally flagged as an outlier .

Variance is the mean of the squared deviations from the mean; standard deviation is its positive square root and is expressed in the original unit. 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 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)

Divide by N N N when your data is the entire group you care about (all 40 students in one section). Divide by n − 1 n - 1 n − 1 when the data is a sample drawn from a larger population. Because n − 1 < n n - 1 < n n − 1 < n , the sample figure is always the larger of the two.

4. Worked Example 2 — standard deviation by hand

Problem. Five students score 6, 8, 9, 10 and 12 on a 15-item quiz. Find the population and sample standard deviations.

Mean. 6 + 8 + 9 + 10 + 12 = 45 6 + 8 + 9 + 10 + 12 = 45 6 + 8 + 9 + 10 + 12 = 45 , so x ˉ = 45 / 5 = 9 \bar{x} = 45/5 = 9 x ˉ = 45/5 = 9 .

Deviations. − 3 , − 1 , 0 , + 1 , + 3 -3,\; -1,\; 0,\; +1,\; +3 − 3 , − 1 , 0 , + 1 , + 3 . They sum to zero, as deviations always must — a free accuracy check.

Squared deviations. 9 , 1 , 0 , 1 , 9 9,\; 1,\; 0,\; 1,\; 9 9 , 1 , 0 , 1 , 9 , whose sum is 20.

Population. σ 2 = 20 / 5 = 4 \sigma^2 = 20/5 = 4 σ 2 = 20/5 = 4 , so σ = 2 \sigma = 2 σ = 2 marks.

Sample. s 2 = 20 / 4 = 5 s^2 = 20/4 = 5 s 2 = 20/4 = 5 , so s = 5 ≈ 2.24 s = \sqrt{5} \approx 2.24 s = 5 ​ ≈ 2.24 marks.

Shifting and scaling

  • Change applied to every value New mean New standard deviation New variance
  • Add a constant c c c mean + c +\;c + c unchanged unchanged
  • Subtract a constant c c c mean − c -\;c − c unchanged unchanged

Multiply by k k k (with k > 0 k > 0 k > 0 ) k × k \times k × mean k × k \times k × SD k 2 × k^2 \times k 2 × variance

So awarding all five students a 5-point bonus moves the mean from 9 to 14 and leaves σ = 2 \sigma = 2 σ = 2 : every score shifted, none spread out. Doubling every score gives a mean of 18, a standard deviation of 4 and a variance of 16.

5. Shape: Skewness and the Empirical Rule

  • Shape Ordering of the centres Typical data
  • Symmetric mean = median = mode heights, well-built test scores
  • Right-skewed (positive) mode < median < mean incomes, house prices, waiting times
  • Left-skewed (negative) mean < median < mode scores on a very easy exam

The mnemonic is simply that the mean chases the tail : whichever side the long tail points to, the mean is dragged that way past the median. Right-skewed (positive) Left-skewed (negative)

^ ^

| .-. | .-.

  • | / \__ | __/ \
  • |/ \____ | ____/ \|
  • +--------------------> +-------------------->
  • Mo Md Mean Mean Md Mo

For a roughly bell-shaped distribution the empirical rule (the 68-95-99.7 rule) says about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three. If a mock exam has mean 78 and standard deviation 6, then about 68% of scores lie between 72 and 84, about 95% between 66 and 90, and about 99.7% between 60 and 96. Because the curve is symmetric, the share above 84 is about ( 100 − 68 ) / 2 = 16 % (100 - 68)/2 = 16\% ( 100 − 68 ) /2 = 16% .

6. Reading the Standard Displays

  • Display Best for How to read it
  • Frequency table Raw counts by class Frequencies sum to n n n ; relative frequency = f / n = f / n = f / n
  • Histogram Continuous data in classes Bars touch ; height = frequency; order is fixed
  • Bar graph Comparing separate categories Bars are separated ; order is arbitrary
  • Line graph Change over time Slope = rate of change; flat = no change
  • Pie chart Parts of a single whole Central angle = relative frequency × 360 ∘ \times\;360^\circ × 36 0 ∘
  • Box plot Spread, symmetry, outliers Five-number summary: min, Q 1 Q_1 Q 1 ​ , median, Q 3 Q_3 Q 3 ​ , max

For the ordered data 12, 15, 17, 18, 21, 24, 26, 30 the median is ( 18 + 21 ) / 2 = 19.5 (18 + 21)/2 = 19.5 ( 18 + 21 ) /2 = 19.5 , Q 1 = ( 15 + 17 ) / 2 = 16 Q_1 = (15 + 17)/2 = 16 Q 1 ​ = ( 15 + 17 ) /2 = 16 , Q 3 = ( 24 + 26 ) / 2 = 25 Q_3 = (24 + 26)/2 = 25 Q 3 ​ = ( 24 + 26 ) /2 = 25 and the IQR is 9. The fences sit at 16 − 13.5 = 2.5 16 - 13.5 = 2.5 16 − 13.5 = 2.5 and 25 + 13.5 = 38.5 25 + 13.5 = 38.5 25 + 13.5 = 38.5 , so nothing here is an outlier. min Q1 Md Q3 max

  • 12 16 19.5 25 30
  • |--------[=====|========]----------|
  • 10 14 18 22 26 30 34

The box holds the middle 50%; a box that is longer on the right with a longer right whisker signals right skew.

7. Common Traps

Not sorting before taking a median. The single most common careless error in this topic.

Taking one middle value when n n n is even. With 6 values you must average the 3rd and 4th, not pick one.

Averaging the frequency column. In a frequency table the frequencies are weights, not data values.

Averaging the midpoints instead of weighting them. In Worked Example 1 that would give 25.5 instead of the correct 27.

Using n n n when the item says "a sample of". The divisor for a sample variance is n − 1 n - 1 n − 1 .

Assuming a bonus changes the spread. Adding a constant shifts the centre and leaves every deviation identical, so the standard deviation is untouched.

Confusing percentage with angle in a pie chart. A 30% slice is 108 ∘ 108^\circ 10 8 ∘ , not 30 ∘ 30^\circ 3 0 ∘ .

Reversing the skew ordering. In a right-skewed set the mean is the largest, because it is the measure the outliers can reach.