LET Secondary Mathematics — Statistics and ProbabilitySummary

A systematic list that records each data value (or class interval) and how often it occurs. For example, if 20 students took a quiz scoring 5, 6, 7, 8, 9, a frequency table shows: Score | Frequency; 5 | 2; 6 | 4; 7 | 6; 8 | 5; 9 | 3. This is the foundation for all other data displays and is especially useful when data repeats or is grouped into intervals.

A visual display using vertical or horizontal bars of equal width to compare distinct categories (discrete data). Example: a bar graph showing the number of students by grade level (Grade 1, 2, 3, etc.). The height of each bar represents the frequency or value for that category. Bars do not touch, emphasizing that categories are separate.

Similar to a bar graph, but used for continuous data grouped into intervals (class intervals). The bars touch, showing that the data flows continuously. Example: a histogram showing the distribution of heights in centimeters with intervals 140–149, 150–159, 160–169. The x-axis shows the intervals; the y-axis shows frequency.

A graph showing change over time, with points connected by line segments. The x-axis usually shows time (hours, days, months, years); the y-axis shows the measured quantity. Example: a line graph showing enrollment in a school over five years, or daily temperature readings.

A circular graph divided into slices, where each slice represents a part of a whole. The entire circle equals 360° and represents 100% of the data. Formula: angle of a slice = (category value ÷ total) × 360°; percentage = (category value ÷ total) × 100%. Example: a pie chart showing how a school's budget is divided among subjects—Math 25%, Science 20%, Language 30%, PE 15%, Arts 10%.

A graph using symbols or pictures to represent data, with a key indicating what each symbol represents. Example: if each picture of a book represents 10 students, then 2.5 book symbols represent 25 students. Pictographs make data accessible to young learners.

The sum of all values divided by the number of values. Formula: Mean = Σ(values) ÷ n. Example: for quiz scores 12, 15, 15, 18, 20, the mean = (12 + 15 + 15 + 18 + 20) ÷ 5 = 80 ÷ 5 = 16. The mean uses every value, making it sensitive to extreme outliers.

The middle value when data is arranged in order. For an odd number of values, it is the center value. For an even number of values, it is the average of the two middle values. Example: data 10, 12, 12, 15, 15 (odd count) has median 12; data 8, 10, 12, 14 (even count) has median (10 + 12) ÷ 2 = 11. The median resists outliers.

The value that appears most frequently in a data set. A set can have one mode (unimodal), two modes (bimodal), multiple modes, or no mode (all values appear once). Example: in 7, 8, 8, 9, 12, the mode is 8 (appears twice). Mode is the only measure of central tendency that works with categorical (non-numeric) data.

The mean of values when each value has an associated weight (frequency or importance). Formula: Weighted Mean = Σ(value × weight) ÷ Σ(weights). Example: a student scores 80 on quizzes (30% weight) and 90 on the final exam (70% weight). Final grade = (0.30 × 80) + (0.70 × 90) = 24 + 63 = 87. This is exactly how term grades are computed in schools.

The difference between the highest and lowest values in a data set. Formula: Range = Maximum – Minimum. Example: for data 7, 8, 8, 9, 12, the range = 12 – 7 = 5. Range is the simplest measure of dispersion but is sensitive to outliers.

The average of the squared deviations from the mean. Step 1: compute the mean; Step 2: find each deviation (value – mean); Step 3: square each deviation; Step 4: average the squares. Formula: Variance = Σ(deviation²) ÷ n. Example: for data 1, 3, 5, 7, 9 with mean 5, deviations are –4, –2, 0, 2, 4; squared: 16, 4, 0, 4, 16, sum 40; variance = 40 ÷ 5 = 8.

The square root of the variance. Formula: SD = √Variance. It has the same units as the original data, making it more intuitive than variance. Example: if variance is 8, then SD = √8 ≈ 2.83. A larger SD indicates greater spread; a smaller SD indicates data is more tightly clustered around the mean.

Quartiles divide ordered data into four equal parts: Q1 (lower quartile, 25th percentile), Q2 (median, 50th percentile), Q3 (upper quartile, 75th percentile). Percentiles generalize this: a score at the 90th percentile beats 90% of the group. The five-number summary (minimum, Q1, median, Q3, maximum) describes both center and spread.

If one choice can be made in m ways and a second independent choice in n ways, the two together can be made in m × n ways. This extends to any number of stages: multiply all the choices. Example: with 4 shirts and 3 pants, there are 4 × 3 = 12 outfits. With 4 choices for first-period subject, 5 for second-period, and 3 for third-period, there are 4 × 5 × 3 = 60 possible schedules.

An arrangement of items where order matters. Formula: P(n, r) = n × (n – 1) × (n – 2) × ... × (n – r + 1) (r factors). For arranging all n items: n! = n × (n – 1) × ... × 1. Example: arranging 4 distinct books on a shelf is P(4, 4) = 4! = 4 × 3 × 2 × 1 = 24 ways. Choosing a first-place, second-place, and third-place winner from 5 contestants is P(5, 3) = 5 × 4 × 3 = 60 ways.

A selection of items where order does not matter. Formula: C(n, r) = P(n, r) ÷ r! = [n × (n – 1) × ... × (n – r + 1)] ÷ r!. Example: choosing 2 people from 5 for a committee is C(5, 2) = (5 × 4) ÷ 2 = 10 ways. Choosing 3 pizza toppings from 8 available is C(8, 3) = (8 × 7 × 6) ÷ (3 × 2 × 1) = 56 ways.

The probability of a single event E is P(E) = (number of favorable outcomes) ÷ (total number of equally likely outcomes). Probability ranges from 0 (impossible) to 1 (certain). Example: drawing a red marble from a bag with 5 red and 3 blue is P(red) = 5 ÷ 8. Complement rule: P(not E) = 1 – P(E). For the red marbles, P(not red) = 1 – 5/8 = 3/8.

For events A and B: P(A or B) = P(A) + P(B) – P(A and B). If A and B are mutually exclusive (cannot both happen), then P(A and B) = 0, so P(A or B) = P(A) + P(B). Example: drawing a king or queen from a deck: P = 4/52 + 4/52 = 8/52 = 2/13 (mutually exclusive). Drawing a red card or a king: P = 26/52 + 4/52 – 2/52 = 28/52 = 7/13 (not mutually exclusive, because two red kings exist).

For independent events A and B: P(A and B) = P(A) × P(B). For dependent events (like drawing without replacement): P(A and B) = P(A) × P(B after A). Example (independent): tossing two coins, P(both heads) = 1/2 × 1/2 = 1/4. Example (dependent): drawing 2 red marbles from a bag of 5 red and 3 blue without replacement: P = 5/8 × 4/7 = 20/56 = 5/14 (after removing one red, 4 red remain of 7).

Data begins as raw values, is organized in a frequency table or graph (presentation), then summarized using mean, median, or mode (central tendency) to describe its center, and finally interpreted in context to answer a question or draw a conclusion. Example: Student scores (80, 85, 88, 92, 95) are presented in a bar graph, the mean is computed as 88, and we conclude that the average student performed well.

Central tendency (mean, median, mode) alone does not reveal how spread out data is. Adding dispersion (range, variance, SD) completes the picture. Two classes may have the same mean score (80) but different SDs (one tight at SD=2, one loose at SD=8), revealing very different levels of consistency. Teachers interpret both together: the mean tells you the center, the SD tells you how consistent the class is.

To compute a probability, you must count favorable outcomes and total outcomes. The fundamental counting principle counts total outcomes. Permutations and combinations are specialized counting techniques for arrangements (order matters) and selections (order doesn't matter). The probability denominator is the total count from the fundamental counting principle.

Simple probability computes P(E) for a single event. Compound events combine two or more simple events using 'and' (multiplication for independent/dependent) or 'or' (addition for mutually exclusive/overlapping). Complex problems chain these: 'Find the probability that in three tosses of a fair coin, we get at least one head.' This uses complement and 'and' together: P(at least one head) = 1 – P(no heads) = 1 – (1/2 × 1/2 × 1/2) = 1 – 1/8 = 7/8.

A frequency table summarizes data into intervals and counts. From this table, you compute the weighted mean (treating frequencies as weights), identify the median and quartiles (by cumulative frequency), and rank data (percentiles). Example: a test-score frequency table shows: 0–20 (5 students), 21–40 (8 students), 41–60 (10 students), 61–80 (6 students), 81–100 (1 student). The median is in the 21–40 range (the 15th data point out of 30, approximately). Weighted mean uses midpoints and frequencies.

DepEd policy requires schools to use appropriate data displays in reports. Bar graphs compare class sections or subjects; line graphs track progress over time; pie charts show budget allocations; histograms display score distributions. Choosing the wrong graph misleads stakeholders. Teachers must select the display that honestly represents the data to parents and supervisors, as required by RA 7836 (Code of Ethics for Professional Teachers) to act with integrity.

When data includes extreme outliers, the mean shifts toward them, becoming unrepresentative. The median resists this shift. Example: family incomes $30k, $35k, $38k, $42k, $500k have mean ≈ $109k (pulled up by the outlier) but median $38k (truly typical). Teachers report the median for skewed data to honestly represent the typical student, upholding RA 7836 standards.

Two coin tosses are independent: the second toss is unaffected by the first. Drawing two marbles without replacement is dependent: the second draw uses an updated count. For independent events, P(A and B) = P(A) × P(B). For dependent events, P(A and B) = P(A) × P(B|A), where P(B|A) is the probability of B given that A occurred. Recognizing this is the key to avoiding calculation errors on the LET.