Normal Distribution Lecture Notes

Question Posed: If we were to poll a large sample population of adult females and plot their respective shoe sizes, what shape would emerge?
- Expected Result: A normal distribution shape, where shoe sizes cluster around a common value.
Special Name: The graphic depiction of shoe sizes of a sample population is called a normal distribution or bell curve.
NOTE: Shoe size is a genetic trait common to the human species; hence, the general population would have mean shoe sizes around a common value with some variation.

Key Characteristics:
- Most data values cluster around a central value, referred to as the mean.
- In a normal distribution: a) Most of the data values cluster around the mean, which is the average of the dataset. b) The relationship among the mean, median, and mode:
  - In a perfectly normal distribution, the mean, median, and mode are equal.
    c) Graph Shape:
  - Peaks: A normal distribution graph has a single peak (unimodal).
  - Symmetry: It is symmetric around the mean. This means:
    - Left half mirrors the right half, which leads to equal areas on both sides of the mean.

Outliers: Data points significantly different from others.
- Types of Skewed Distributions:
- Left-skewed (Mode > Median > Mean): The tail on the left side is longer; mean and median are less than mode.
- Right-skewed (Mean > Median > Mode): The tail on the right side is longer; mean and median are greater than mode.
- Symmetric Distribution (Mean = Median = Mode): Equal distribution of numbers on both sides.

Low Variation: Data points are close to the mean.
Moderate Variation: Varying distances from the mean, but still fairly close.
High Variation: Large spread of data points in comparison to the mean.

Considerations:
- Normal:
- a) Heights of adult men
- c) Times of runners at the Olympic marathon
- e) I.Q. scores of the general population
- Skewed:
- b) Scores on a very easy test (likely many perfect scores)
- d) I.Q. scores of Nobel prize scientists (may be skewed due to exceptionally high scores).

Data Value Distribution: Standard statistical results within a normal distribution include:
- Approximately 68% of data falls within 1 standard deviation of the mean.
- Approximately 95% of data falls within 2 standard deviations of the mean.
- Approximately 99.5% of data values fall within 3 standard deviations of the mean.
Standard Normal Distribution:
- Empirical Rule for Standard Deviations:
  - Within 1 std dev: ( ext{Area} ext{ to the left} ext{ of } ext{mean} = 34 ext{%}), so total = 68% for both sides.
  - Within 2 std devs: ( ext{Area} = 95 ext{%}).
  - Within 3 std devs: ( ext{Area} = 99.7 ext{%}).

Given: A dataset with a mean of 45 and a standard deviation of 8.3.
- Calculations Needed:
1. +1 standard deviation from the mean: 45 + 8.3 = 53.3
2. +3 standard deviations from the mean: 45 + 3 imes 8.3 = 70.9
3. -1 standard deviation from the mean: 45 - 8.3 = 36.7
4. -2 standard deviations from the mean: 45 - 2 imes 8.3 = 28.4
- Note: Sketch a normal curve for each distribution and label the x-axis firmly at intervals of one, two, and three standard deviations from the mean.

New Data Set: Normal distribution with mean = 5.1 and standard deviation = 0.9.
- Find Percent of Data:
1. Between 6.0 and 6.9: Requires calculation based on Z-scores.
2. Greater than 6.9: Also derived from Z-scores.
3. Between 4.2 and 6.0: Factor in standard deviations.
4. Less than 4.2: Similar analysis.
5. Less than 5.1: Direct conclusion as it is the mean.
6. Between 4.2 and 5.1: Calculate using established percentages.

Normal Distribution of Test Scores:
- Given information: Mean = 76, Standard deviation = 10.
- Count Based on Group Size (230 students):
  a. Students above 96 score: Requires finding Z-score.
  b. Students below 66 score: Calculate using Z-score method.
  c. Within one standard deviation of the mean: 68% of students.
- To find this, compute the number of students as: 0.68 imes 230 = 156.4 ext{ (approx. 156 students)}.

Frequency Distribution: Shows the frequency of outcomes within a given situation.
Common Patterns: Can vary between normal, skewed, or bimodal distributions.

Task: Predict the shape and reasoning for distributions:
- (a) Heights of Toronto Raptors basketball team: Likely normal distribution due to genetic similarities.
- (b) Cost of 1 L of gas: Likely normal distribution due to market price adjustments.
- (c) Masses of players on Canadian Olympic hockey teams: Likely normal distribution due to similar training and genetics.

Normal Distribution: Height of adult males
Skewed Distribution: Income levels
Bimodal Distribution: Test scores for students with different academic preparation.