Question 1175995: The following is a frequency distribution of ages of residents of a senior citizens' home in New York. Fill in the midpoints, then calculate the mean and the standard deviation. Round to two decimal positions. You may assume the data represents a population.
Ages of Residents:59 - 65, 66 - 72, 73 - 79, 80 - 86
Midpoints:62,69,76,83
Frequency:2,3,4,1
Mean (μ)=
Standard deviation (σ) =
Round the standard deviation up to 2 decimal places if necessary.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's calculate the mean and standard deviation for the given frequency distribution.
**1. Calculate the Mean (μ)**
* We use the formula: μ = Σ(midpoint * frequency) / Σfrequency
* Σ(midpoint * frequency) = (62 * 2) + (69 * 3) + (76 * 4) + (83 * 1) = 124 + 207 + 304 + 83 = 718
* Σfrequency = 2 + 3 + 4 + 1 = 10
* μ = 718 / 10 = 71.80
**2. Calculate the Standard Deviation (σ)**
* We use the formula: σ = √[Σ(frequency * (midpoint - μ)²) / Σfrequency]
* First, calculate (midpoint - μ)² for each midpoint:
* (62 - 71.8)² = (-9.8)² = 96.04
* (69 - 71.8)² = (-2.8)² = 7.84
* (76 - 71.8)² = (4.2)² = 17.64
* (83 - 71.8)² = (11.2)² = 125.44
* Next, multiply each result by its corresponding frequency:
* 96.04 * 2 = 192.08
* 7.84 * 3 = 23.52
* 17.64 * 4 = 70.56
* 125.44 * 1 = 125.44
* Sum these values: 192.08 + 23.52 + 70.56 + 125.44 = 411.6
* Divide by the total frequency: 411.6 / 10 = 41.16
* Take the square root: √41.16 ≈ 6.415606
* Rounding to two decimal places: σ ≈ 6.42
**Answers:**
* Mean (μ) = 71.80
* Standard Deviation (σ) = 6.42
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