Statistical distributions show how data values are spread and help us understand patterns in large datasets.
Bell-shaped curve
68-95-99.7 Rule:
68% within 1 standard deviation
95% within 2 standard deviations
99.7% within 3 standard deviations
Uniform: All values equally likely
Skewed: Tail on one side
Bimodal: Two peaks
Symmetric: Mirror image around mean
Mean = Median = Mode
Standard deviation (σ): Controls width
Area under curve = 1
Z = (X - μ) / σ
Where X = data value, μ = mean, σ = standard deviation
Given: Test scores normally distributed
Mean (μ) = 75, Standard deviation (σ) = 10
Find Z-score for X = 85:
Z = (85 - 75) / 10 = 10/10 = 1
Z = 1: 1 standard deviation above mean
Z = 0: Equal to the mean
Z = -1: 1 standard deviation below mean
Question 1: Heights are normally distributed with mean 170cm and standard deviation 8cm. What is the Z-score for a height of 186cm?
Question 2: In a normal distribution, what percentage of data falls within 2 standard deviations of the mean?
Question 3: If test scores have mean = 80 and σ = 5, what score corresponds to Z = -1.5?