LET Elementary Mathematics — Statistics and ProbabilityRevision Notes

Formulas

Exam Tips

• When a question shows a graph, read the axis labels and scale FIRST before answering — the trick is almost always in the scale.
• If asked which graph is MOST APPROPRIATE: bar = categories; line = trend over time; pie = parts of one whole; histogram = distribution of continuous grouped data.
• Pie-chart angle items are quick: just multiply the fraction by 360. Practice the mental shortcut: 25% = 90°, 50% = 180°, 10% = 36°.
• For pictograph items, always multiply the number of symbols by the key value before computing percentages or totals.

Key Points

• A frequency table lists values or class intervals alongside their counts (frequencies) and is the foundation for most other displays.
• Bar graph: compares distinct categories (discrete data); bars do NOT touch.
• Histogram: shows the distribution of continuous data grouped into intervals; bars DO touch.
• Line graph: displays change over time (trends); best for continuous data across equal time intervals.
• Pie (circle) graph: shows parts of a whole; the entire circle = 360° = 100% of the data.
• Pictograph: uses a symbol where each symbol represents a fixed count; always check the key.
• Choosing the WRONG graph type is itself a tested item — match the graph to the data and purpose.
• Misleading graphs: a vertical axis that does not start at zero, or unequal intervals, can make small differences look large.
• Pie-chart angle formula: angle = (category frequency ÷ total) × 360°
• Pie-chart percentage formula: percentage = (category frequency ÷ total) × 100
• Before drawing a conclusion from any graph, identify the axes, their units, and the scale.

Definitions

Section Title

Data Presentation: Graphs, Tables, and Charts

Common Mistakes

• Using a pie graph to compare two separate totals — pie graphs only show parts of ONE whole.
• Confusing a bar graph (categories, bars apart) with a histogram (continuous intervals, bars touching).
• Forgetting that a pictograph's value depends on the KEY — each symbol may represent more than one unit.
• Computing a sector angle using percentage instead of the ratio (f ÷ N), leading to a result > 360° when percentages are misapplied.
• Misreading a scale where the y-axis does not start at zero, and treating a visual gap as a large numerical difference.

Formulas

Exam Tips

• ALWAYS check: Is the data ordered? If not, reorder before finding the median.
• Watch for outliers in the data set — if there is one extreme value, the median is likely the better descriptor, and the LET may ask you to identify which measure is most appropriate.
• Weighted mean shortcut: if weights are given as percentages, convert to decimals (e.g., 30% → 0.30) and multiply each score by its decimal weight; sum the products — that IS the weighted mean.
• Mode items are often the fastest to answer — scan the data for repeats first.
• For frequency table mean items, set up a column for (x × f) to avoid arithmetic errors.

Key Points

• The mean is the arithmetic average: sum all values, divide by the count. It uses EVERY value, so it is pulled toward extreme scores (outliers).
• The median is the MIDDLE value of an ORDERED data set. It resists outliers, making it best for skewed distributions like income data.
• For an ODD count of values, the median is the single middle value.
• For an EVEN count of values, the median is the AVERAGE of the two middle values.
• The mode is the value that occurs MOST OFTEN. A data set can be unimodal, bimodal, multimodal, or have NO mode.
• The mode is the ONLY measure of central tendency that can be used for CATEGORICAL (nominal) data (e.g., favorite subject: Math, Science, Filipino).
• Weighted mean: each value is multiplied by its weight, the products are summed, and the total is divided by the sum of the weights.
• Weighted mean is EXACTLY how DepEd computes term grades combining quizzes (Written Work), performance tasks, and quarterly exams.
• When data is in a frequency table, the mean = Σ(value × frequency) ÷ Σfrequency.
• Skewed right (positive skew): mean > median > mode; skewed left (negative skew): mean < median < mode.

Definitions

Section Title

Measures of Central Tendency: Mean, Median, and Mode

Common Mistakes

• Forgetting to ORDER the data before finding the median — the most common median error.
• For an even number of values, reporting one of the two middle values instead of their average.
• Confusing 'no mode' with a mode of zero — if no value repeats, there is simply NO MODE.
• Using the mean for categorical data (e.g., computing the 'average' favorite subject) instead of the mode.
• In weighted mean problems, dividing by the number of components instead of the sum of the weights when weights are not given as decimals.

Formulas

Exam Tips

• For SD problems on the LET, follow the exact five-step procedure: mean → deviations → square deviations → average squares (variance) → square root. Set up a table to keep organized.
• If two classes have the SAME MEAN, compare their SDs — the class with the SMALLER SD performed more consistently. This is a classic item type.
• Remember: SD can never be negative. If you get a negative answer, you made an arithmetic error.
• Five-number summary questions: list Min, Q1, Median, Q3, Max in order — any one of these five can be asked.
• Percentile items: 'scored at the 85th percentile' means 85% of test takers scored BELOW that student, not that the student scored 85%.

Key Points

• Dispersion (spread) describes how far data values are from each other and from the center.
• Range = highest value − lowest value. It is the simplest measure but is very sensitive to outliers.
• Variance = the mean of the squared deviations from the mean; always non-negative.
• Standard Deviation (SD) = √Variance. It is expressed in the SAME UNITS as the data.
• A LARGER SD means data points are MORE SPREAD OUT (less consistent); a SMALLER SD means data points are CLOSER to the mean (more consistent).
• Two classes can have the same mean but very different SDs — the SD is what reveals which class performed more consistently.
• Population variance uses n (divide by the total count); sample variance uses n-1. The LET generally uses population formulas.
• Steps to compute SD: (1) Find the mean. (2) Subtract mean from each value (deviations). (3) Square each deviation. (4) Average the squared deviations (= variance). (5) Take the square root.
• Quartiles split ordered data into FOUR equal parts: Q1 (25th percentile), Q2 = median (50th percentile), Q3 (75th percentile).
• Interquartile Range (IQR) = Q3 − Q1; measures the spread of the middle 50% of data and is resistant to outliers.
• Five-number summary: Minimum, Q1, Median (Q2), Q3, Maximum — the basis of a box-and-whisker plot.
• Percentile rank: a score at the 90th percentile means 90% of the group scored below that value.

Definitions

Section Title

Measures of Dispersion: Range, Variance, and Standard Deviation

Common Mistakes

• Forgetting to SQUARE the deviations before averaging them — computing the mean of unsigned deviations always gives zero.
• Taking the square root of the DEVIATIONS instead of the variance — always square first, average, then take the square root at the end.
• Confusing variance and standard deviation in an answer — variance is in squared units (e.g., points²); SD is in original units (points).
• Computing the range using the frequency instead of the actual score values.
• Misidentifying Q1 and Q3 when n is even — use the median of the lower half for Q1 and the median of the upper half for Q3.

Formulas

Exam Tips

• The ONE deciding question: 'Does swapping the positions/identities of two chosen members create a NEW, DIFFERENT outcome?' YES → Permutation. NO → Combination.
• Quick C(n,r) computation tip: write out n × (n−1) × ... for r factors in the numerator, then divide by r!. You rarely need to compute large factorials fully.
• Memorize: C(n,1) = n; C(n,n) = 1; C(n,0) = 1. These often appear in quick-answer items.
• For FCP items with restrictions (e.g., 'the first digit cannot be 0'), handle the RESTRICTED position FIRST, then multiply the remaining choices.
• Common LET item pattern: 'In how many ways can a teacher select 2 pupils from a class of 30?' → C(30,2) = (30×29)÷2 = 435.

Key Points

• Fundamental Counting Principle (FCP): if Event 1 can occur in m ways and Event 2 can occur in n ways, both together can occur in m × n ways. Extend by multiplying all stages.
• Example of FCP: 4 shirts × 3 pants × 2 pairs of shoes = 24 possible outfits.
• A PERMUTATION counts arrangements where ORDER MATTERS (the sequence is important).
• A COMBINATION counts selections where ORDER DOES NOT MATTER (only who/what is chosen matters).
• Key test cue: if swapping two chosen items creates a DIFFERENT outcome → PERMUTATION; if it creates the SAME outcome → COMBINATION.
• Factorial: n! = n × (n−1) × (n−2) × ... × 2 × 1. By definition, 0! = 1.
• P(n, r) = n! ÷ (n − r)! — arrangements of r items chosen from n distinct items.
• C(n, r) = n! ÷ [r! × (n − r)!] — selections of r items from n distinct items (order irrelevant).
• Relationship: C(n, r) = P(n, r) ÷ r! — combinations are permutations with the repeated arrangements removed.
• Permutation examples: seating arrangements, ranking finalists, forming passwords, arranging books on a shelf.
• Combination examples: forming committees, selecting a team, choosing topics to study, drawing lottery numbers.

Definitions

Section Title

Fundamental Counting Principle, Permutations, and Combinations

Common Mistakes

• Using P(n,r) when the question asks for a committee or group (order does not matter) — always ask 'does the order/rank matter?'
• Using C(n,r) when the question assigns specific roles or rankings — if titles like president, secretary, treasurer are given, use P.
• Forgetting that 0! = 1, leading to errors in the denominator of C(n,r) when r = n.
• In FCP problems, multiplying when options are NOT independent — check that each choice is truly separate.
• Cancelling factorials incorrectly: P(5,3) = 5! ÷ 2! = 120 ÷ 2 = 60, NOT 5!/3! — always subtract r from n in the denominator.

Formulas

Exam Tips

• The single most important decision: Is the question asking for AND (both) → MULTIPLY, or OR (either) → ADD? Read the question carefully.
• For 'at least one' problems, ALWAYS use the complement: P(at least one) = 1 − P(none). This saves time and prevents errors.
• Check your answer: Is the probability between 0 and 1? If not, recheck your denominator.
• For two-dice problems: the total sample space is always 6 × 6 = 36. List favorable pairs systematically.
• For card problems: a standard deck has 52 cards — 4 suits of 13 cards each (Ace, 2–10, Jack, Queen, King).
• The phrase 'without replacement' is the signal for DEPENDENT events — reduce both numerator and denominator appropriately for subsequent draws.

Key Points

• Probability of event E: P(E) = (number of favorable outcomes) ÷ (total number of equally likely outcomes).
• Probability is ALWAYS between 0 (impossible) and 1 (certain), inclusive.
• The sum of the probabilities of ALL possible outcomes equals 1.
• Complement Rule: P(not E) = 1 − P(E). Use this for 'at least one' problems — it is almost always faster.
• A SIMPLE event is a single outcome or a group of outcomes from one experiment.
• A COMPOUND event combines two or more simple events using 'AND' or 'OR'.
• MUTUALLY EXCLUSIVE events: two events that CANNOT BOTH occur at the same time (e.g., rolling a 3 AND a 5 on one die). P(A and B) = 0.
• INDEPENDENT events: the outcome of one event does NOT affect the other (e.g., tossing two separate coins).
• DEPENDENT events: the outcome of the first event DOES affect the second (e.g., drawing two cards without replacement).
• Addition Rule (OR): P(A or B) = P(A) + P(B) − P(A and B).
• For MUTUALLY EXCLUSIVE events: P(A or B) = P(A) + P(B) (the overlap is zero).
• Multiplication Rule (AND) — Independent: P(A and B) = P(A) × P(B).
• Multiplication Rule (AND) — Dependent: P(A and B) = P(A) × P(B|A), where P(B|A) uses UPDATED counts after the first event.
• The DENOMINATOR is the most common error source — it must equal the TOTAL number of equally likely outcomes in the sample space.

Definitions

Section Title

Probability of Simple and Compound Events

Common Mistakes

• Using the WRONG DENOMINATOR — the denominator must be the TOTAL number of equally likely outcomes in the sample space at the time of the event.
• Adding probabilities for 'AND' events instead of multiplying (and vice versa).
• Forgetting to subtract the overlap in P(A or B) when events are NOT mutually exclusive.
• In without-replacement problems, keeping the denominator the same for both draws instead of reducing it by 1.
• Treating the complement of 'at least one' as 'exactly one' instead of 'none at all.'