LET Secondary Assessment of Learning — Statistics, Grading and Interpreting Assessment ResultsDetailed Explanation

Measures of Central Tendency: Mean, Median, and Mode

A measure of central tendency is a single number that best represents an entire set of scores — it tells you where the 'center' of the distribution lies. There are three measures, each with a specific strength and appropriate use case. **MEAN** — The arithmetic average. Formula: Mean = (Sum of all scores) / (Number of scores), often written as X̄ = ΣX / n. The mean uses every score in the calculation, which makes it the most mathematically precise measure. However, this also means it is heavily influenced by extreme scores (outliers). If one pupil scores very high or very low compared to the rest, the mean shifts toward that extreme value. For this reason, the mean is BEST USED when the distribution is roughly symmetrical and has no extreme outliers. **MEDIAN** — The middle score when all scores are arranged from lowest to highest. If n is odd, the median is the single middle score. If n is even, it is the average of the two middle scores. The median is exactly the 50th percentile (P50). Because it depends only on the position of scores — not their actual values — the median is NOT affected by outliers. When a few pupils score extremely high or extremely low and pull the mean away from the bulk of scores, the MEDIAN better represents the typical performance. Use the median for skewed distributions. **MODE** — The score that appears most frequently. A distribution can be unimodal (one mode), bimodal (two modes), multimodal (many modes), or have no mode (when all scores occur equally). The mode is the ONLY measure of central tendency that can be used with nominal (categorical) data, such as the most common subject chosen by pupils. It requires no calculation — just counting frequency. **Worked Example:** Seven Grade 4 pupils scored 82, 85, 85, 88, 90, 92, 94 in a Science quiz. - Sum = 82+85+85+88+90+92+94 = 616; n = 7 - Mean = 616 / 7 = 88 - Arranged in order: 82, 85, 85, [88], 90, 92, 94 → Median = 88 (4th score, the middle) - 85 appears twice, all others once → Mode = 85 In this example, all three measures are close together, indicating a roughly symmetrical distribution. When they diverge significantly, it signals skewness — the topic of a later concept in this chapter.

Measures of Variability: Range and Standard Deviation

Knowing where scores center (central tendency) is only half the picture. You also need to know how SPREAD OUT the scores are. This is called variability or dispersion. Two classes with the same mean of 80 could be very different: Class A has scores ranging from 78 to 82 (very tight), while Class B has scores from 50 to 100 (very spread). Variability statistics capture this difference. **RANGE** — The simplest measure of spread. Formula: Range = Highest Score - Lowest Score. Quick and easy to compute, but it depends on only two extreme scores and ignores everything in between. A single extreme outlier can inflate the range drastically, making it an unstable measure. **STANDARD DEVIATION (SD)** — The most important and reliable measure of variability. Conceptually, the SD tells you the AVERAGE DISTANCE of scores from the mean. If the SD is small, most scores cluster close to the mean (the group is HOMOGENEOUS). If the SD is large, scores are widely scattered (the group is HETEROGENEOUS). Formula for Population SD: SD = √[Σ(X - X̄)² / n] Step-by-step process: 1. Find the mean (X̄) 2. Subtract the mean from each score: (X - X̄) — these are called deviations 3. Square each deviation: (X - X̄)² 4. Find the average of the squared deviations: Σ(X - X̄)² / n — this is the VARIANCE 5. Take the square root of the variance — this is the SD **VARIANCE** — The square of the standard deviation. If SD = 2.83, then Variance = 8. The LET may ask for either. **Worked Example:** Scores: 4, 6, 8, 10, 12. Mean = (4+6+8+10+12)/5 = 40/5 = 8. Deviations: 4-8=-4, 6-8=-2, 8-8=0, 10-8=+2, 12-8=+4 Squared deviations: 16, 4, 0, 4, 16 Sum of squared deviations = 16+4+0+4+16 = 40 Variance = 40/5 = 8 SD = √8 ≈ 2.83 Interpretation: On average, scores sit about 2.83 points away from the mean of 8. **Classroom Meaning of SD:** - Small SD (e.g., 2–3): Scores tightly packed around the mean → homogeneous class → relatively similar ability levels - Large SD (e.g., 10–15): Scores widely spread → heterogeneous class → wide ability range → teacher needs differentiated instruction **INTERQUARTILE RANGE (IQR)** = Q3 - Q1. This captures the spread of the MIDDLE 50% of scores and, like the median, resists outliers. It is used alongside the median when the distribution is skewed.

Percentiles, Quartiles, and Deciles

Percentiles, quartiles, and deciles are measures of RELATIVE POSITION. Unlike the mean and SD which describe the group as a whole, these statistics tell you WHERE ONE SCORE FALLS in relation to all others. **PERCENTILE (P)** — Percentiles divide an ordered distribution into 100 equal parts. A score at the 90th percentile (P90) means the student scored AS WELL AS OR BETTER THAN 90% of the group. This is the CRITICAL LET trap: a percentile rank of 90 does NOT mean the student answered 90% of the items correctly. It is a RELATIVE (norm-referenced) position, not an absolute score. Example: Maria has a percentile rank of 85 in the national achievement test. This means she scored as well as or better than 85% of all test-takers. She might have answered only 60% of items correctly — the percentile says nothing about that. **QUARTILES (Q)** — Quartiles divide the ordered distribution into FOUR equal parts: - Q1 (First Quartile) = P25: 25% of scores fall AT OR BELOW this point - Q2 (Second Quartile) = P50: the MEDIAN; 50% of scores fall at or below this point - Q3 (Third Quartile) = P75: 75% of scores fall at or below this point **DECILES (D)** — Deciles divide the distribution into TEN equal parts (D1 through D9): - D1 = P10, D2 = P20, D3 = P30 … D5 = P50 = Q2 = Median … D9 = P90 **KEY CHAIN OF EQUIVALENCIES (frequently tested on the LET):** Q2 = P50 = D5 = Median Q1 = P25; Q3 = P75; D1 = P10; D9 = P90 **INTERQUARTILE RANGE (IQR)** = Q3 - Q1: The spread of the middle 50% of scores. Because it excludes the top and bottom 25% of scores, it is resistant to outliers. Used alongside the median for skewed distributions. **How to Find Quartiles from Raw Data:** 1. Arrange all scores from lowest to highest. 2. Find the median (Q2 = P50). 3. Q1 is the median of the LOWER half of scores (scores below Q2). 4. Q3 is the median of the UPPER half of scores (scores above Q2). Example: Scores: 60, 65, 70, 75, 80, 85, 90, 95 - Q2 = median = (75+80)/2 = 77.5 - Lower half: 60, 65, 70, 75 → Q1 = (65+70)/2 = 67.5 - Upper half: 80, 85, 90, 95 → Q3 = (85+90)/2 = 87.5 - IQR = 87.5 - 67.5 = 20

The Normal Curve

The NORMAL CURVE (also called the bell curve or Gaussian distribution) is one of the most important concepts in educational measurement. It is a theoretical, symmetrical, bell-shaped distribution that describes how many human characteristics (height, weight, intelligence, and standardized test scores for large populations) tend to distribute themselves naturally. **KEY PROPERTIES OF THE NORMAL CURVE:** 1. **Symmetrical** — the left half is a mirror image of the right half. 2. **Mean = Median = Mode** — all three measures of central tendency coincide at the CENTER of the curve. 3. **Unimodal** — it has only one peak (one mode). 4. **Asymptotic** — the tails extend infinitely toward both ends but never touch the baseline (x-axis). 5. **Total area = 1.00** (or 100%) — all scores fall under the curve. 6. **Defined by two parameters:** the mean (determines the center/location) and the standard deviation (determines the width/spread). **THE 68-95-99.7 RULE (Empirical Rule):** This is the most important fact about the normal curve for the LET: - About **68%** of scores fall within **±1 SD** of the mean (between mean-1SD and mean+1SD) - About **95%** of scores fall within **±2 SD** of the mean - About **99.7%** of scores fall within **±3 SD** of the mean **Practical Example:** A standardized Math test has a mean of 80 and SD of 5. - 68% of pupils scored between 75 and 85 (80±5) - 95% of pupils scored between 70 and 90 (80±10) - 99.7% of pupils scored between 65 and 95 (80±15) - A pupil scoring 90 is at +2 SD, above approximately 97.5% of test-takers **BELOW THE NORMAL CURVE — AREA BREAKDOWN:** From center outward (one side): 34.13% between 0 and +1SD; 13.59% between +1 and +2SD; 2.14% between +2 and +3SD; 0.13% beyond +3SD. This mirrors on the negative side. These percentages connect directly to z-scores and percentile ranks. **WHY IT MATTERS IN THE CLASSROOM:** Under a normal distribution, most pupils perform in the middle range (near the mean), and fewer pupils score extremely high or extremely low. In contrast, if almost ALL pupils score very high or very low, the distribution is NOT normal — it is skewed. Recognizing this helps teachers evaluate their tests: a good classroom test for a diverse group should produce roughly normal scores. However, after good teaching, you EXPECT a negatively skewed distribution (most pupils scoring high).

Skewness: The Classic LET Trap

Skewness describes the ASYMMETRY of a distribution — how much it deviates from the perfectly symmetrical normal curve. This is THE single most frequently tested and most frequently MISUNDERSTOOD concept in this chapter on the LET. The key rule: **SKEWNESS IS NAMED AFTER THE TAIL, NOT THE HUMP.** **THREE TYPES OF DISTRIBUTIONS:** 1. **NORMAL (SYMMETRICAL):** The tail on the right equals the tail on the left. The bulk of scores is in the center. Mean = Median = Mode. Indicates BALANCED test difficulty. 2. **POSITIVELY SKEWED (Skewed to the RIGHT):** The long tail points toward the HIGH end (right). The bulk of scores PILES UP at the LOW end (left). This means MOST PUPILS SCORED LOW. The test was DIFFICULT (or the class performed poorly). Relationship of measures: **Mean > Median > Mode** (the mean is dragged toward the high tail; the mode is at the hump, which is at the low end). 3. **NEGATIVELY SKEWED (Skewed to the LEFT):** The long tail points toward the LOW end (left). The bulk of scores PILES UP at the HIGH end (right). This means MOST PUPILS SCORED HIGH. The test was EASY (or the class performed well). Relationship of measures: **Mean < Median < Mode** (the mean is dragged toward the low tail; the mode is at the hump, which is at the high end). **LOGICAL REASONING (to avoid memorizing blindly):** - The MEAN chases the tail (it is pulled toward extreme scores). - The MODE is at the hump (the tallest point — where most scores are). - The MEDIAN is always between the mean and mode. - In positive skew: tail → high scores → mean goes high; hump → low scores → mode is low. So Mean > Median > Mode. - In negative skew: tail → low scores → mean goes low; hump → high scores → mode is high. So Mean < Median < Mode. **LET SCENARIO STRATEGY:** - If the scenario says 'most pupils scored very LOW / few scored high' → POSITIVELY SKEWED → difficult test - If the scenario says 'most pupils scored very HIGH / few scored low' → NEGATIVELY SKEWED → easy test - The direction of skew = direction of the TAIL, not where the bulk is. **AFTER GOOD TEACHING:** A teacher who has taught well expects a NEGATIVELY SKEWED distribution (most pupils score high, demonstrating mastery). A test given at the very start of a unit (before teaching) often produces a POSITIVELY SKEWED distribution.

Standard Scores: z-scores and T-scores

A raw score by itself is meaningless without context. A score of 40 could be excellent (in a 50-item test where the mean is 35) or poor (in a 50-item test where the mean is 48). STANDARD SCORES solve this problem by expressing each raw score in terms of its distance from the mean, measured in standard deviation units. This places scores from different tests on a COMMON, COMPARABLE SCALE. **THE z-SCORE** Formula: z = (X - X̄) / SD Where: X = raw score; X̄ = mean of the distribution; SD = standard deviation The z-score tells you HOW MANY STANDARD DEVIATIONS a score lies ABOVE or BELOW the mean: - A POSITIVE z-score: the raw score is ABOVE the mean - A NEGATIVE z-score: the raw score is BELOW the mean - z = 0: the raw score IS the mean - z = +1: one SD above the mean - z = -1: one SD below the mean The z-distribution has a MEAN of 0 and a STANDARD DEVIATION of 1. **Worked Example:** Class mean = 80, SD = 4. - Ana scored 88: z = (88-80)/4 = 8/4 = +2.0 (2 SDs above the mean; roughly top 2%) - Ben scored 76: z = (76-80)/4 = -4/4 = -1.0 (1 SD below the mean) - Cora scored 80: z = (80-80)/4 = 0/4 = 0 (exactly at the mean) **THE T-SCORE** Negative z-scores confuse parents and administrators, so scores are often converted to a more user-friendly scale. The T-score formula transforms z-scores: Formula: T = 50 + 10z The T-distribution has a MEAN of 50 and a STANDARD DEVIATION of 10. There are no negative T-scores in practical ranges (a z of -5 would give T=0, which is essentially impossible). Using the same example: - Ana: z=+2.0 → T = 50 + 10(2.0) = 50 + 20 = 70 - Ben: z=-1.0 → T = 50 + 10(-1.0) = 50 - 10 = 40 - Cora: z=0 → T = 50 + 10(0) = 50 **COMPARING PERFORMANCE ACROSS SUBJECTS (The Classic LET Computation):** Carlo scored 85 in Math (class mean 80, SD 5) and 88 in English (class mean 86, SD 4). In which subject did he perform better RELATIVE TO HIS CLASS? - Math: z = (85-80)/5 = 5/5 = +1.0 - English: z = (88-86)/4 = 2/4 = +0.5 Despite the LOWER raw score in Math, Carlo performed BETTER IN MATH (+1.0 SD vs. +0.5 SD above his respective class means). THIS IS THE KEY USE OF z-SCORES: comparing relative performance across subjects, not comparing raw scores. **Reverse computation (from z to raw score):** If you know z = +1.5, mean = 70, SD = 8, find X: X = X̄ + z(SD) = 70 + 1.5(8) = 70 + 12 = 82

Correlation

A CORRELATION COEFFICIENT (r) describes the RELATIONSHIP between two sets of scores or two variables. In educational assessment, correlations are used to establish reliability, validity, and to understand relationships between pupil characteristics and performance. **TWO PIECES OF INFORMATION IN EVERY CORRELATION:** 1. **DIRECTION (the sign of r):** - **POSITIVE CORRELATION (+):** Both variables move in the SAME direction. When one goes up, the other goes up; when one goes down, the other goes down. Example: Study time and test scores (more study hours → higher grades). - **NEGATIVE CORRELATION (-):** Variables move in OPPOSITE directions. When one goes up, the other goes down. Example: Number of absences and final grades (more absences → lower grades). - **ZERO or near-zero correlation:** No meaningful linear relationship between the variables. 2. **STRENGTH (the magnitude/absolute value of r):** r ranges from -1.00 to +1.00. The CLOSER the absolute value is to 1.00, the STRONGER the relationship. A sign tells you direction; magnitude tells you strength. **Interpretive Scale (conceptual guide used in education):** - |r| = 0.00 to 0.20: Negligible (no practical relationship) - |r| = 0.21 to 0.40: Low (weak relationship) - |r| = 0.41 to 0.60: Moderate - |r| = 0.61 to 0.80: High (strong relationship) - |r| = 0.81 to 1.00: Very High (very strong relationship) **CRITICAL RULE:** r = -0.85 is STRONGER than r = +0.60, because |-0.85| = 0.85 > |+0.60| = 0.60. The negative sign means they move in opposite directions — it does NOT mean the relationship is weaker. **APPLICATIONS IN ASSESSMENT:** - **Test-retest reliability** is a correlation between scores from the same test given twice; a high positive r (e.g., +0.90) means the test is consistent/reliable. - **Criterion-related validity** is a correlation between test scores and a criterion (e.g., grades); a high positive r means the test predicts the criterion well. - **Item analysis** sometimes uses point-biserial correlation to check whether each item discriminates between high and low performers. **CRITICAL GUARD RAIL — CORRELATION IS NOT CAUSATION:** A high correlation between two variables does NOT mean one causes the other. Example: There may be a high positive correlation between the number of books in a home and a child's academic performance, but having more books does not CAUSE high performance — both may be caused by a third variable (educated, affluent parents). Teachers must be careful not to imply causation from correlation data when reporting to parents or stakeholders.

DepEd K-12 Grading System: DepEd Order No. 8, s. 2015

The DEPED ORDER NO. 8, s. 2015 (Policy Guidelines on Classroom Assessment for the K to 12 Basic Education Program) is the official legal basis for how Filipino elementary teachers compute and report grades. As a licensed professional teacher under RA 7836, you are bound to implement this policy faithfully. Understanding it thoroughly is essential both for the LET and for actual classroom practice. **PHILOSOPHY:** The K-12 grading system is STANDARDS-BASED and COMPETENCY-BASED. Grades reflect the degree to which a learner has met the standards and competencies identified in the curriculum, NOT just performance on tests alone. Assessment is holistic, encompassing products, performances, and written assessments. **THREE COMPONENTS OF THE QUARTERLY GRADE:** Every subject's quarterly grade is built from three components: 1. **WRITTEN WORK (WW)** — Quizzes, short tests, unit tests, essays, written exercises, and other written outputs that demonstrate knowledge and understanding of the competencies. 2. **PERFORMANCE TASKS (PT)** — Tasks that require pupils to demonstrate what they know and can DO in real-life situations: demonstrations, projects, experiments, oral presentations, portfolios, role-plays, laboratory activities. Performance Tasks carry the HIGHEST or among the highest weights in ALL subject areas, reflecting the K-12 program's learner-centered, performance-based thrust. 3. **QUARTERLY ASSESSMENT (QA)** — The written examination (or performance-based assessment) administered at the end of each quarter to determine how well learners have mastered the quarter's competencies. **COMPONENT WEIGHTS BY SUBJECT (Grades 1-10):** | Subject Area | Written Work | Performance Tasks | Quarterly Assessment | |---|---|---|---| | Languages (Filipino, English), Araling Panlipunan, EsP | 30% | 50% | 20% | | Mathematics, Science | 40% | 40% | 20% | | MAPEH, EPP/TLE | 20% | 60% | 20% | Note: The Quarterly Assessment carries 20% in ALL subject areas for Grades 1-10. Performance Tasks always carries a heavy weight, emphasizing application over rote recall. **HOW TO COMPUTE THE QUARTERLY GRADE — STEP BY STEP:** Step 1: Compute each component's PERCENTAGE SCORE (PS) PS = (Total Raw Score / Highest Possible Raw Score) × 100 Example: WW total raw score = 45 out of 60 possible → PS = (45/60) × 100 = 75 Step 2: Multiply each PS by the component's WEIGHT to get WEIGHTED SCORES Using Math subject weights (WW=40%, PT=40%, QA=20%): - WW: 75 × 0.40 = 30.0 - PT: PS_PT × 0.40 - QA: PS_QA × 0.20 Step 3: ADD the three weighted scores → INITIAL GRADE Initial Grade = Weighted WW + Weighted PT + Weighted QA Step 4: TRANSMUTE the Initial Grade using the TRANSMUTATION TABLE The transmutation table converts the initial grade (which has 60 as its minimum realistic value) to the reported grade scale: - Initial Grade of 100 → Transmuted Grade of 100 - Initial Grade of 60 → Transmuted Grade of 75 (MINIMUM PASSING) - Grades between 60 and 100 are transmuted proportionally - The LOWEST TRANSMUTED GRADE reported on the report card is 60 (for learners who did not participate at all) **GRADE DESCRIPTORS:** | Transmuted Grade | Descriptor | |---|---| | 90-100 | Outstanding | | 85-89 | Very Satisfactory | | 80-84 | Satisfactory | | 75-79 | Fairly Satisfactory | | Below 75 | Did Not Meet Expectations | **PASSING GRADE: 75** (Fairly Satisfactory and above — the pupil moves to the next grade level) **FINAL GRADE:** Average of the FOUR quarterly grades in each subject. **PROMOTION RULES:** A learner who receives a Final Grade of 75 or above in all subjects is PROMOTED to the next grade level. A learner who fails (below 75) in not more than two subjects is given remediation during the school year; if they still fail after remediation, they are retained.

Even when the three measures are close, we check which is largest: if Mean > Median ≥ Mode, it is still positive skew. The extreme score of 98 is the outlier causing the mild positive pull. In practice, Teacher would use the MEDIAN (82) as the more representative measure since the mean is skewed by the one very high score. This problem integrates central tendency, mode, and skewness — a common LET multi-part item.

Despite Maria having a higher raw score (85 vs. 78), Juan performed better RELATIVE TO HIS CLASS. This is the fundamental purpose of standard scores: to compare performance across groups with different means and SDs. On the LET, the answer is always about z-scores for relative comparison, not raw scores. Note also that both pupils scored above their class means (positive z), but Juan outperformed his class by a greater margin.

Always use the correct weights for the subject. For Filipino (a Language subject), PT carries 50% — not 40%. A common LET trap is using Math/Science weights (40%-40%-20%) for Language subjects. The Performance Task score of 88 heavily influenced the final grade (it carries 50% weight), pulling the initial grade higher than the Written Work and QA scores alone would suggest. This demonstrates why Performance Tasks are so important in the K-12 grading system.

This problem integrates quartiles, percentiles, and IQR. Part (b) reinforces the chain Q3=P75 — a frequently tested equivalency. Part (c) reinforces the meaning of Q1=P25: 25% of the group falls at or below that point. The IQR of 25 (from 70 to 95) captures the spread of the middle 50% of the class and is not affected by the extreme score of 60 or 100.

Part (a) and (b) directly apply the 68-95-99.7 rule. Part (c) requires dividing the tail percentage (5%) in half because the curve is symmetric. Part (d) connects z-scores to approximate percentile ranks under the normal curve. For the LET, knowing that z=+1≈P84, z=+2≈P97.5, z=-1≈P16 is sufficient for most items without a z-table.