1. Draw the Center: The Mean ($\mu$) is always the peak of the hill.
2. Step Out: Standard Deviation ($\sigma$) is the size of the "step" to the left and right.
3. The 68-95-99.7 Rule: Use these fixed percentages for every normal distribution.
4. The Z-Score: $Z = (Value - \text{Mean}) / \text{Standard Deviation}$.
A group of 1,000 students take a math test. The Mean score is 75 ($\mu = 75$) and the Standard Deviation is 5 ($\sigma = 5$).
Label the Hill: Draw a bell curve and label the values for 1, 2, and 3 standard deviations away from the center.
Apply the Rule:
A) How many students scored between 70 and 80? (68%)
B) How many students scored between 65 and 85? (95%)
C) How many students scored above 85?
If the Standard Deviation was 10 instead of 5... would the hill be taller or wider? Would there be more or fewer students scoring exactly 75?
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Calculate the Z-score for these individual data points.
The High Achiever: A student scores a 90 on the test ($\mu=75, \sigma=5$). What is their Z-score?
The Struggle: A student scores a 65. What is their Z-score?
Comparing Different Groups:
A Grape bunch weighs 600g ($\mu=500, \sigma=50$).
An Apple weighs 250g ($\mu=200, \sigma=20$).
Which fruit is more "exceptional" in its group? (Compare Z-scores).
The Height of the Tribes: The average height of men in a tribe is 180cm with a standard deviation of 10cm.
What percentage of men are taller than 200cm?
If you know the Z-score you want, you can find the Value ($x$):
$x = \mu + (Z \cdot \sigma)$.
Task: You want to be in the top 0.15% of the class. This means you need a Z-score of +3. If the mean is 75 and the deviation is 5... what score do you need to get?
Objective: Explain the Bell Curve to a younger student using a handful of beans or LEGOs.
The Activity:
1. Sort a large bag of mixed blocks by size or color.
2. Line them up to show that most are one "common" type, and only a few are "weird" types.
The Lesson: "God loves 'Normal' because it's how we work together. But He also made the 'Tails' to show us that He can do anything. We are one body with many different members."
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