Chapter 10 - Probability - Normal Distributions/Area Under Curve/Z - Scores

Question 1

How do you graph a box and whiskers plot? (4)

Answer 1

1. Box itself is drawn from Q1 to Q3
2. Line drawn inside box = median
3. Whiskers: Left line drawn from Q1 to smallest NON-OUTLIER & right line drawn from Q3 to largest NON-OUTLIER
4. Points beyond whiskers = outliers

Question 2

What is a relative frequency histogram? (4)

Answer 2

1. Bar graph whose height represents relative frequency of data in interval
2. Y‑axis is NOT number of people but percent/fraction of people
3. X‑Axis shows intervals of data (12-14, 14-16, 16-18)
4. All bar heights (frequencies) together add up to 1 (or 100%), so area under histogram represents whole dataset

Question 3

Definition of Mean, Median, Mode, Q1, Q3 & Interquartile Range (6)

Answer 3

• Mean: Average
• Median: Middle Number
• Mode: Most Frequent Number (If all numbers have same frequency, there’s no mode)
• Q1: Median of Lower Half (25% of Data Fall Under)
• Q3: Median of Upper Half (75% of Data Fall Under)
• Interquartile Range: (Q3 - Q1), Focuses on Middle 50% of Data

Question 4

How do you calculate the outliers? (6)

Answer 4

1. Order data (values from smallest → largest)
2. Identify Median, Split Data in 2 Halves
3. Find Q1 (Median of Lower Half) & Q3 (Median of Upper Half)
4. IQR = Q3 − Q1, 1.5 × IQR
5. Q1 − (1.5 × IQR) = Lower Fence (any value < LF is an outlier)
6. Q3 − (1.5 × IQR) = Upper Fence (any value > LF is an outlier)

Question 5

Definition of Standard Deviation (4)

Answer 5

• Spread of Data: Describes how far data values fall around mean (measures how wide/tight distribution is)
• Small SD: Data tightly clustered near mean, Large SD: Data values spread far from mean
• “Standard Deviation Away”: SD Value Calculator Provides = Typical Distance from Mean, 2 SD = 2 x typical distance (above/below mean)

Question 6

How do you find standard deviation in the calculator? (3)

Answer 6

1. STAT, ENTER, 1 (Edit), Type Data into L1 (Frequencies in L2 if Provided)
2. STAT, CALC, 1 (1‑Var Stats), ENTER, Type L1 (2nd → 1), Type L2 into Frequency List (if needed/provided)
3. Read Population Standard Deviation Value (σx)

Question 7

What are the three different types of distributions and how do you accurately account for the data differently? (3)

Answer 7

1. Symmetrical Distribution (Bell Curve, Median = Peak): Mean = Median = Mode
   - Use Mean for Center & Standard Deviation for Spread
2. Skewed Right: Tail extends right, Most Data on Left
   - Use Median for Center & IQR for Spread (b/c mean pulled upward by Rt tail)
3. Skewed Left: Tail Extends Left, Most Data on Right
   - Use Median for Center & IQR for Spread (b/c mean pulled downward by Lt tail)

Question 8

How do you find the center and spread of a normal curve? (3)

Answer 8

1. Normal Curve: Symmetric bell curve where mean is at peak (L/R sides are roughly equal)
2. Center: Median (also equal to mean)
3. Spread: USE STANDARD DEVIATION
   - A normal bell curve ALWAYS contains about 3 SDs of data on each side
   (mean to right edge = 3 SDs, mean to left edge = 3 SDs)
   - Total width of curve contains about 6 Standard Deviations
   - To estimate Standard Deviation: SD ≈ (Max − Min) ÷ 6

Question 9

Definition of area under curve & how to calculate it (6)

Answer 9

1. Shaded region under a normal curve b/w two x-values = probability (percent/proportion) of data falls in interval
2. Area is represented as a percent or a decimal, and under the whole normal curve equals 1 (or 100% of data)
3. μ (mu): Mean of Distribution, Center of Curve, 0 in Standard Normal Curve
4. Standard Normal Distribution:
   - Even Bell Curve
   - Mean (μ) = 0 on X Axis
   - Standard Deviation “Typical Distance” (σ) = 1 on X Axis
   - Curve is ALWAYS used for z-scores & Normal CDF on Calculator
5. X-Axis:
   - Spacing b/w x-values is measured in standard deviation units (each step = 1 SD)
   - Shows z-scores: tells you how many SDs a value is from mean (left of 0 = negative z (below mean), right of 0 = positive x (above mean))

Question 10

What do you do if the provided normal curve is NOT a standard normal curve
(mean is not 0 and standard deviation is not 1)? (3)

Answer 10

• Still use Normal CDF but must replace mean + SD w/ ones given in problem
• In normalcdf, enter:
  Lower bound = A
  Upper bound = B
  Mean = (given μ)
  Standard deviation = (given σ)
• The calculator will then compute the correct probability for that specific curve.

Question 11

What does a Z-Score Mean? (1/2/3/3)

Answer 11

1. Number of SDs a value is above/below mean
2. Standardizes All Scores:
   
   • Converts raw scores to z scores into “standard deviation units,” placing all datasets on same scale
   • This allows fair comparison across different tests, units, or distributions to see someone’s true standing.
3. Connection to Percentiles:
   
   • Each z‑score corresponds to a percentile based on area under normal
     curve to LEFT of z‑score
   • This percentile shows what percent of the population the score is higher than (higher percentile = beats more people)
   • Percentile is calculated through “Normal CDF”
4. Interpretation of Numbers:
   - Negative z → percentile below 50%
   - z = 0 (mean) → 50th percentile
   - Positive z → percentile above 50%
   Ex: Different SAT test dates have different difficulty levels, so raw scores from
   each month are not directly comparable. A 1300 in March might represent a very
   different performance than a 1300 in October. To make admissions fair, all SAT scores from different months (different datasets) are converted into z‑scores. This standardizes every student’s score into “standard deviation units,” placing all test dates on the same common scale. Once standardized, colleges compare students by their z‑scores (and resulting percentiles), not by raw SAT scores. This shows each student’s true standing relative to everyone else, regardless of which month’s test they took.

Question 12

Definition of percentile and how to calculate it (4)

Answer 12

1. What percent of the population is at or BELOW given score (84th percentile = higher than 84% of population)
2. Represents area under normal curve to LEFT of that z‑score (or raw score)
3. Calculated through use of z scores (each z score corresponds to percentile based on location of x axis of curve and therefore the area under curve to the left of it)
4. Calculator:
   - Use Normal CDF to find the area to the LEFT of a score
   - 2nd → VARS → 2: normalcdf
   - Lower bound = -1E99, upper bound = (your score/z-score), Mean = given μ, Standard deviation = given σ, if no mean/SD are provided, assume standard normal curve (μ = 0, σ = 1)
   - Output: Calculator returns the percentile as a decimal

Question 13

What are all the different ways to calculate components relating to area under curve?

Answer 13

1. Converting a Score to Z Score
   - Plug Score into: Z Score = (Your Score - Mean)/Standard Deviation
   - Output = z-score whose area to the LEFT equals that percentile
2. Find what score across different datasets will have the same z-score
   - One of Data Sets (Given Original Score): Convert Original Raw Score/Interval into Z-Scores (Repeat 2X for Intervals)
   - Other Data Set (No Score Given): Plug that z-score into equation & solve for raw score
3. Finding Probability b/w Two Values (A to B):
   - Use the calculator’s Normal CDF to find probability
   - 2nd → VARS → 2: normalcdf, Lower bound = A & Upper bound = B
   - Output is probability (percent of data) between A and B
4. Finding the Percentile of a Z Score/Probability BELOW a Score
   - Use Normal CDF to find probability from -∞ to that z-score
   - 2nd → VARS → 2: normalcdf, Lower bound = -1E99 & Upper bound = Z-Score
   - Output is probability (percent of data) of people BELOW the Z-Score
5. Probability of Being ABOVE a Z-Score
   - Use Normal CDF to find probability from z-score to +∞
   - 2nd → VARS → 2: normalcdf, Lower bound = Z-Score & Upper bound = 1E99
   - Output is probability (percent of data) of people BELOW Z-Score
6. Finding the Z-Score at a Given Percentile
   - Use INVNORM (inverse normal) to find percentile (area = percentile)
   - 2nd → VARS → 3: invNorm, Area = Percentile (As a Decimal)

Question 14

How do you calculate the probability of finding a data value between two intervals in a curve? (3)

Answer 14

• Use the calculator’s Normal CDF to find probability
• 2nd → VARS → 2: normalcdf, Lower bound = A & Upper bound = B
• Output is probability (percent of data) between A and B

Question 15

How do you calculate the percentile of a z score? (3)

Answer 15

• Use Normal CDF to find probability from -∞ to that z-score
• 2nd → VARS → 2: normalcdf, Lower bound = -1E99 & Upper bound = Z-Score
• Output is probability (percent of data) of people BELOW the Z-Score

Question 16

How do you calculate the probability of being ABOVE a Z-Score? (3)

Answer 16

• Use Normal CDF to find probability from z-score to +∞
• 2nd → VARS → 2: normalcdf, Lower bound = Z-Score & Upper bound = 1E99
• Output is probability (percent of data) of people BELOW Z-Score

Question 17

How do you find the Z-Score at a Given Percentile? (3)

Answer 17

• Use INVNORM (inverse normal) to find percentile (area = percentile)
• 2nd → VARS → 3: invNorm, Area = Percentile (As a Decimal)
• Output = z-score whose area to the LEFT equals that percentile

Question 18

How do you convert raw scores into z‑scores?

Answer 18

Plug into : Z-Score = (Your Score - Mean)/Standard Deviation

Question 19

How do you find what score across different data sets will have the same z score?

Answer 19

• One of Data Sets (Given Original Score): Convert Original Raw Score/Interval into Z-Scores (Repeat 2X for Intervals)
• Other Data Set (No Score Given): Plug that z-score into equation & solve for raw score