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Mastering the Standard Normal Table A Step-by-Step Guide to Reading Z-Scores and Probabilities NotebookLM.
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What is a Z-Score? • A Z-score (standard score) describes a specific position within a distribution. • It measures exactly how many standard deviations a data point is from the mean (0). - Z-scores + Z-scores (Below Mean) (Above Mean) -1 -3 -2 1 3 Mean (0) 2 Key Concept • Positive Z (+): Value is above average. • Negative Z (-): Value is below average. • Z = 0: Value is exactly average. NotebookLM.
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Why Do We Use a Z-Table? • The Normal Distribution is a density curve with a total area of 1 (100%). • The formula calculates position; the table translates that position into a usable probability. 0.0 0.1 0.2 0.3 0.5 Determine Probability. aa@a oooa ooon OOOU O. 6 0.00 .5000 .5040 .5100 .5160 .5200 .5288 .5378 0.01 .5040 .5080 .5160 .5240 .5260 .5300 .5420 0.02 .5080 .5110 .5184 .5220 .5286 .5317 .5463 .5398 .5400 .5488 .5612 .5254 .5308 .5398 NotebookLM.
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The Anatomy of a Z-TabIe • The table is a coordinate grid designed for Z-scores up to two decimal places. Columns: 2nd Decimal (Hundredths) Rows: Integer + 1st Decimal 1.0 1.0 .00 .8413 .8413 .8413 .8423 .8423 .8425 .8523 .8522 .8526 .8420 .01 .8438 .8438 .8438 .8454 .847t .8572 .8604 .8118 .8128 .8134 .02 .8461 .8461 .8461 .8561 .03 .8478 .8478 .8498 .8716 .04 . 8474 .8496 .8712 .8720 .05 .8498 .8608 .8654 .8738 .06 .8520 .8520 .8653 .8745 .07 .8524 .8640 .8686 .8758 Body: Probability / Area to the Left 786 . bbU1 .8511 .8538 .8532 .8140 .8569 .8507 .8056 .8149 . BBIU .8548 .8870 .9367 .8342 .884B .9868 .9371 .9376 .9350 .8845 .8970 .9378 .9380 .9351 .8806 .5831 .5800 .5394 .5326 .08 .8538 .8648 .8686 .8719 .5778 .5826 .5835 .5847 .5396 .5363 .8554 .8672 .8710 .8747 .5754 .5789 .5813 .5867 .5394 .5398 NotebookLM.
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How to Read the Coordinates • To find a value, decompose the Z-score into two parts: 1. Part A: First two digits lookup the Row. 2. Part B: Last digit looks up the Column. Horizontal Lookup (Column) e--1023 Vertical Lookup (Row) tqotebookLM.
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Example 1: Small Z- Score (z = 0.0 0.1 0.2 .5000 .5398 .5793 .5040 .5438 .5832 .5080 .5478 .5871 .5120 .5517 .5910 .5160 .5557 .05 .5199 .5596 Intersection .5948 .5987 0.13) Step 1: Find row 0.1 in the left column. Step 2: Find column .03 in the top row. Step 3: Trace to the intersection. Result: 0.5517 (55.17% of data is below this score). NotebookLM.
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Modern Academic Precision Example 2: Medium Z-Score (z = .8 85 .8 08 .9 82 .9 36 z 1.0 1.1 1.3 1.4 .00 .8413 .8643 .9032 .9192 .01 .8438 .8665 .9049 .9207 .02 .8461 .8686 . 88 .9066 .9222 3 .04 .8508 .8729 .9099 .9251 .05 .8531 .8749 .9115 .9265 1.23) Step 1: Identify the first part: 1.2. Step 2: Identify the precision part: .03. Step 3: Find where they meet. Result: 0.8907 (Roughly 89% of population is below). NotebookLM.
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Example 3: Larger Z-Score (z = 2.15) z 2.0 2.1 2.2 .00 .9772 .9821 .9861 .01 .9778 .9826 .9864 .02 .9783 .9830 .9868 .03 .9788 .9834 .9871 .04 .9793 .05 .9798 .983 0.9842 .9875 Intersection Step 1: Locate row 2.1 on the left. Step 2: Locate column .05 on the top. Step 3: Read the intersection value. Result: 0.9842. Note: As Z increases, the probability approaches 1. tqotebookLM.
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Example 4: Outlier Z-Score (z = Helvetica Now Display z 3.0 3.1 3.2 3.3 3.4 .00 .9987 .9990 .9993 .9994 .9995 .01 .9987 .9990 .9993 .9995 .9996 .02 .9987 .9991 .9994 .9996 .9997 .03 Almost 100% .9995 .9997 .9998 .9996 .9998 .9999 3.22) Step 1 : Scroll down to row 3.2. Step 2: Move across to column .02. Step 3: Identify the value. Result: 0.9994 Significance: An extremely rare outlier. NotebookLM.
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Interpretation: Area to the Left < 1.23) = • The Golden Rule: The Table always gives the area to the LEFT of the Z-score. • This represents the cumulative probability up to that point. 0.8907 Table Value (0.8907) z = 1.23 x tqotebookLM.
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Interpretation: Area to the Right • Sometimes we need the proportion above a score. • Since Total Area = 1, use subtraction. • Formula: Area to Right = $1 - (Table value) $1 - o. 157 = 0.2843 O z =0.57 z tqotebookLM.
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Interpretation: Area Between Two Scores • To find the proportion between two values: • Logic: Take the larger total area and subtract the smaller total x area. z = 0.62 z = 1.6 z Area(1.6) [Large] - Area(0.62) [Smalt] = Target Area 0.9452 - 0.7324 = 0.2128 NotebookLM.
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Common Mistakes to Avoid x Row/CoI Confusion Remember: Whole number + ISt decimal = Row. Only 2nd decimal = Column. Direction Error The table never gives the Area to the Right directly. Subtract from 1. 1.23 Rounding Errors Always round Z-scores to 2 decimal places before lookup. tqotebookLM.
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Summary & Key Takeaways A. Left Value Calculate Z • The Table. split Z (Row/Col) Find Intersection (Table Value) Adjust Direction B. Right Value Big - Small C. Between • Z-Scores: Measure distance from mean. • Maps scores to probabilities (Left Area). • Flexibility: Use logic to find Less Than, Greater Than, or Between. tqotebookLM.
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O Practice is Key • Mastery of the Z-table comes from repetition. • Remember: Any normal distribution can be standardized and solved using this single tool. Lecture Complete. tqotebookLM.