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Zorluk: OrtaMeasures of Central Tendency for Grouped Data

An environmental research station recorded the daily particulate matter concentration (in μg/m3\mu\text{g/m}^3) near an urban center over a period of 5050 days. The observations are summarized in the table below:

Particulate Matter (μg/m3\mu\text{g/m}^3)Number of Days (ff)
20−2920 - 2966
30−3930 - 391010
40−4940 - 491515
50−5950 - 591111
60−6960 - 6988

Find the estimated mean particulate matter concentration, in μg/m3\mu\text{g/m}^3, for the 50-day period.

Cevap: 45.5 µg/m³

Cevap

The estimated mean particulate matter concentration is 45.5 μg/m345.5\text{ }\mu\text{g/m}^3.
To calculate the mean of grouped data, determine the midpoint (xx) of each class interval: 24.524.5, 34.534.5, 44.544.5, 54.554.5, and 64.564.5. Multiply each midpoint by its frequency (ff) to get the products 147147, 345345, 667.5667.5, 599.5599.5, and 516516. The sum of these products (∑fx\sum fx) is 22752275. Dividing ∑fx\sum fx by the total frequency (∑f=50\sum f = 50) yields the estimated mean of 45.5 μg/m345.5\text{ }\mu\text{g/m}^3.

Adım Adım Çözüm

1
Calculate the class midpoints (xx) for each class interval
Midpoints are 24.524.5, 34.534.5, 44.544.5, 54.554.5, and 64.564.5.
Grouped data uses the midpoint of each interval to represent all values falling within that interval.
2
Compute the product of frequency and midpoint (fxfx) for each class
6×24.5=1476 \times 24.5 = 147, 10×34.5=34510 \times 34.5 = 345, 15×44.5=667.515 \times 44.5 = 667.5, 11×54.5=599.511 \times 54.5 = 599.5, 8×64.5=5168 \times 64.5 = 516.
This determines the estimated sum of data values within each group.
3
Sum all products ∑fx\sum fx and total frequency ∑f\sum f
∑fx=147+345+667.5+599.5+516=2275\sum fx = 147 + 345 + 667.5 + 599.5 + 516 = 2275 and ∑f=50\sum f = 50.
These totals are required for the mean formula.
4
Calculate the mean concentration using xˉ=∑fx∑f\bar{x} = \frac{\sum fx}{\sum f}
xˉ=227550=45.5\bar{x} = \frac{2275}{50} = 45.5.
Dividing the total estimated sum by the total number of days gives the estimated mean.

Anahtar Kavram

Calculation of Mean for Grouped Frequency Data
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Konular

Measures of Central Tendency for Grouped Data

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