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Normal Distribution Practice: Calculate Median, Standard Deviation, and Quartiles
Mathematics
Grade 11 (Senior High School)
Question Content
Consider the normal distribution shown to the right, and assume that P and P' are the two points of inflection of the curve. Complete parts (a) through (d) below: (a) Find the median M of the distribution. (b) Find the standard deviation σ of the distribution. (c) Find the third quartile Q₃ of the distribution. (d) Find the first quartile Q₁ of the distribution. The normal curve has a peak at 80 in, and inflection points at 73 in and 87 in.
Correct Answer
(a) M = 80 in; (b) σ = 7 in; (c) Q₃ = 85 in; (d) Q₁ = 75 in
Detailed Solution Steps
1
Step 1: Solve part (a) - Find the median. In a normal distribution, the mean, median, and mode are all equal and located at the center (peak) of the curve. The peak is at 80 in, so the median M = 80 in.
2
Step 2: Solve part (b) - Find the standard deviation. The points of inflection of a normal distribution occur at μ - σ and μ + σ, where μ is the mean (equal to the median 80 in). Given one inflection point is 80 + σ = 87 in, solve for σ: σ = 87 - 80 = 7 in. We can verify with the other inflection point: 80 - 7 = 73 in, which matches the given value.
3
Step 3: Solve part (c) - Find the third quartile Q₃. For a normal distribution, the third quartile is approximately μ + 0.675σ. Substitute μ = 80 and σ = 7: Q₃ = 80 + (0.675 * 7) = 80 + 4.725 ≈ 85 in (rounded to the nearest inch).
4
Step 4: Solve part (d) - Find the first quartile Q₁. For a normal distribution, the first quartile is approximately μ - 0.675σ. Substitute μ = 80 and σ = 7: Q₁ = 80 - (0.675 * 7) = 80 - 4.725 ≈ 75 in (rounded to the nearest inch).
Knowledge Points Involved
1
Properties of Normal Distribution (Center)
In a normal distribution, the mean, median, and mode are identical and located at the center (peak) of the symmetric bell curve. This means the center value represents all three measures of central tendency.
2
Inflection Points of Normal Distribution
A normal curve changes concavity at its inflection points, which are exactly one standard deviation (σ) away from the mean (μ), located at μ - σ and μ + σ. This relationship is used to calculate the standard deviation from given inflection point values.
3
Quartiles in Normal Distribution
For a normal distribution, the first quartile (Q₁) is approximately μ - 0.675σ, and the third quartile (Q₃) is approximately μ + 0.675σ. These values mark the 25th and 75th percentiles of the distribution, respectively, dividing the data into four equal parts.
4
Standard Deviation in Normal Distribution
Standard deviation (σ) is a measure of spread in a normal distribution. It quantifies how far data points are from the mean, with about 68% of data falling within μ ± σ, 95% within μ ± 2σ, and 99.7% within μ ± 3σ.
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