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Zorluk: Çok zorMeasures of Central Tendency (Mean, Median, Mode)

A dataset consists of 2525 distinct integers arranged in increasing order. The mean of all 2525 integers is 5252. The mean of the smallest 1212 integers is 3030, and the mean of the largest 1212 integers is 7070.

If 55 additional numbers, each equal to the median of the original dataset, are added to the dataset, what is the mean of the new set of 3030 numbers?

Cevap: 60

Cevap

The mean of the new set of 30 numbers is 60.
The sum of all 25 numbers is 25×52=130025 \times 52 = 1300. Since the numbers are ordered, the median is the 13th value. The 12 smallest numbers sum to 12×30=36012 \times 30 = 360 and the 12 largest sum to 12×70=84012 \times 70 = 840. The sum of these 24 numbers is 360+840=1200360 + 840 = 1200, so the 13th number (the median) must be 1300−1200=1001300 - 1200 = 100. Adding 5 copies of 100 increases the sum to 1300+500=18001300 + 500 = 1800 across 3030 numbers. The new mean is 180030=60\frac{1800}{30} = 60.

Adım Adım Çözüm

1
Calculate the total sum of the original dataset of 25 numbers.
Sum = 25×52=130025 \times 52 = 1300.
The mean multiplied by the number of elements gives the total sum.
2
Calculate the combined sum of the 12 smallest and 12 largest integers.
Sum of 12 smallest = 12×30=36012 \times 30 = 360; Sum of 12 largest = 12×70=84012 \times 70 = 840; Total = 360+840=1200360 + 840 = 1200.
The 25 numbers consist of the 12 smallest, the 1 median (13th element), and the 12 largest.
3
Determine the value of the median.
Median = 1300−1200=1001300 - 1200 = 100.
Subtracting the sum of the 24 non-median values from the total sum yields the 13th element, which is the median.
4
Find the sum and count of the modified dataset.
New Sum = 1300+5(100)=18001300 + 5(100) = 1800; New Count = 25+5=3025 + 5 = 30.
Adding 5 numbers each equal to 100 increases the sum by 500 and the count by 5.
5
Calculate the mean of the new dataset.
New Mean = 180030=60\frac{1800}{30} = 60.
Divide the new total sum by the new total count of numbers.

Anahtar Kavram

Relationship between Mean, Median, and Total Sum in Partitioned Datasets
Tahmini Süre:2m 30s
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Konular

Measures of Central Tendency (Mean, Median, Mode)

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