Welcome to the live session of Statistics and Average Part I. The questions discussed on October 13, 2017 during the live session follow the embedded video. The video link next to each question will take you to the start of the part of the video where that specific question is discussed. Click the video embedded below to watch the entire session. Concept covered in this session, simple average (arithmetic mean), weighted average (weighted mean), range, median, and mode.
The average age of students of class A comprising 20 students is 20 years and that of students of classes A and B together is 16 years. If class B comprises 40 students, what is the average age of students of class B?
The average salary of a graduating class of 30 students is $8420 per month and that of another class comprising 20 students is $8438 per month. What is the average monthly salary of the students of the two classes taken together?
The average of 5 positive integers is 40. The largest value among the 5 is 44. What is the minimum value possible for the smallest of these 5 numbers?
Aaron was asked to calculate the arithmetic mean of ten positive integers each of which had two digits. By mistake, he interchanged the two digits, say a and b, in one of these ten integers (ab). As a result, his answer for the arithmetic mean was 2.7 more than what it should have been. Find (b - a).
The range of the weight of men in a team is 12 kg, and that of the women in the team is 10 kg. What is the range of the weight of the team if the lightest man in the team is 3 kg lighter than the heaviest woman in the team?
Consider 11 distinct integers whose median is 90 and range is 60. The median for the five smallest integers is 65. What is the maximum range for the five largest integers?
The arithmetic mean of 5 positive integers a, b, c, d, and e is 22, and a < b < c < d < e. If e is 40, what is the greatest possible value of the median of the 5 integers? What is the least possible value of the median of the 5 integers?
A set contains the following observations : a, ak, ak^{2} where a, k > 0. For what value of 'k' will the median of the set be greater than the arithmetic mean?
Average or arithmetic mean is one of the measures of central tendency of a data set. It is also referred to as the typical value of a data set.
The formula used to compute average is Average = \\frac{\text{Sum of all terms}}{\text{Number of terms}} )
Depending upon what information is given, the same formula can be used to compute the sum of all terms and the number of terms.
If the average of a data set and the number of terms is given, Sum of terms = Average × (Number of terms)
If the average of a data set and the sum of terms is given, Number of terms = \\frac{\text{Sum of all terms}}{\text{Average of the terms}} )
Weighted average is computing the average after taking into consideration the difference in degrees of importance of numbers in a set. For instance, if we are asked to compute the average marks scored by students of two classes A and B. If the number of students in the two classes are different, the average that we compute is known as weighted average or weighted mean.
Let's roll with the above example. Let Class A have n1 students and let the average marks scored by students of Class A be a1. Let the number of students in Class B be n2 and the average marks scored by students of Class B be a2. What is the average marks scored by students of the two classes together?
We can compute the weighted average by drawing a table as shown below or by using the formula to compute weighted average.
Class | Number of Students | Average Marks | Sum of Marks |
---|---|---|---|
Class A | n1 | a1 | n1 × a1 |
Class B | n2 | a2 | n2 × a2 |
Both classes together | n1 + n2 | weighted average | n1 × a1 + n2 × a2 |
Therefore, weighted average = \\frac{\text{Sum of marks of students of classes A and B}}{\text{Total number of students in A and B}})
This is the formula used to compute weighted average = \\frac{\text{(n1 × a1) + (n2 × a2)}}{\text{n1 + n2}})
Note: If the numbers of students in the two classes are the same, n1 = n2, say n.
The formula basically becomes \\frac{\text{(n × a1) + (n × a2)}}{\text{n + n}}) = \\frac{\text{a1 + a2}}{\text{2}} )
i.e., If the weightages are equal, the weighted average is the same as the simple average.
Median is the middle point in a data set. The median is the 50th percentile value when talk about a large data set. Median is the value the separates the lower half of a population from the higher half of the population.
If a data set has an odd number of data points, arrange the elements in ascending order. The middle value is the median.
For example, let a data set has the following numbers: 18, 5, 103, 37, -11
The data set has 5 elements. Arrange the numbers in ascending order: -11, 5, 18, 37, 103.
The third number is the median. 18 is the third term and the middle value. Hence, 18 is the median.
If a data set has an even number of elements, arrange the elements in ascending order. The simple average of the middle two values is the median.
For example, let a data set has the following numbers: 7, 91, 28, 56
The data set has 4 elements. Even number of terms. Arrange the numbers in ascending order: 7, 28, 56, 91
The median is the arithmetic mean of the middle two terms viz., average of 28 and 56.
Median = \\frac{\text{28 + 56}}{\text{2}} )
Mode is the value that appears the most number of times in a data set. It is the value that has the highest probability of being selected if a value is selected at random from the given data set.
For example, let a data set have the following numbers: 4, 1, 9, 3, 1, 0, 1
The value that appears most frequently is 1. Hence, 1 is the mode of this data set.
Range of a data set is the difference between the largest and the smallest value in the data set. Range is one of the measures of dispersion of the elements of a data set. Greater the range, higher the dispersion.
For example, let a data set have the following numbers: 8, 109, -5, 217.
The largest number is 217 and the smallest number is -5.
The range of the set = largest number - smallest number = 217 − (-5) = 222
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