Formulas
Average = $\left(\dfrac{Sum\;of \;observations}{Number\; of\; observations}\right)$
Number of Observations = Total ÷ Average
Sum of Observation = Average × Number of Observation.
Weighted average
Weighted Average= $\dfrac{Sum \;of \;Weighted \;Terms}{Total \; Number\; of \;Terms}$
Average Speed:
Average Speed = $\left(\dfrac{Total\; Distance}{Total\;Time}\right)$
Suppose a man covers a certain distance at x kmph and an equal distance at y kmph
.Then, the average speed during the whole journey is = $\left(\dfrac{2xy}{x+y}\right)$
Average
Average refers to the sum of numbers divided by n. Also called the mean average.
Sums of data divided by the number of items in the data will give the mean average.
Average can be calculated only for similar quantities and not for dissimilar quantities.
Average of height and weight cannot be calculated. It should either be average height of all students or average weight of all students.
Also Known As: Central tendency. A measure of the middle value of the data set.
Average = $\left(\dfrac{Sum\;of \;observations}{Number\; of\; observations}\right)$
Example:
Find the average of 5, 7, 6, 8, 4, 9
Solution:
Sum of the given number = 5 + 7 + 6 + 8 + 4 + 9 = 39
Number of events = 6
Therefore, average of numbers = $\dfrac{39}{6}$
= $\dfrac{13}{2}$
= 6.5
Exercise:
Sara’s average marks in Maths Unit Tests are = $\dfrac{85 + 89 + 98}{3}$
= $\dfrac{272}{3}$
= 90.6
Thus, Sara’s average marks in Maths Unit Tests = 90.6 %.
Using an Average to Find a Number of Observations:
Sometimes you will be asked to find a number by using a given average.
Number of Observations = $\left(\dfrac{Sum\;of \;observations}{Average}\right)$
Example:
Total height of a class is 1300 cm. If the average height of a class is 65 cm, find the number of students in the class.
Solution:
Total height of a class = 1300 cm.
Average = 65 cm.
No. of students = $\left(\dfrac{Total}{Average}\right)$
= $\left(\dfrac{1300}{65}\right)$
= 20
Therefore, number of students in the class = 20
Exercise:
Total height of a class = 1500 cm.
Average = 50 cm.
No. of students = Total ÷ Average
= 1500 ÷ 50
= 30
Therefore, number of students in the class = 20
Using an Average to Find Sum of Observations:
Sometimes you will be asked to find sum by using a given average.
Sum of Observation = Average × Number of Observation.
Example:
The average consumption of wheat by a family is 33 kgs in three months. If there are 15 members in the family, find the total consumption for three months.
Solution:
Average = 33 kgs.
No. of members = 15
Total = Average × No. of members.
= 33 × 15
= 495 kg.
Therefore, the total consumption of wheat for 3 months is 495 kg.
Exercise:
Average = 45 kgs.
No. of members = 15
Total = Average × No. of members.
= 45 × 15
= 675 kg.
Therefore, the total consumption of wheat for 3 months is 675 kg.
Weighted average
The average between two sets of numbers is closer to the set with more numbers.
Weighted Average= $\dfrac{Sum \;of \;Weighted \;Terms}{Total \; Number\; of \;Terms}$
Example:
A class of 25 students took a science test. 10 students had an average score of 80. The other students had an average score of 60.Find weighted average.
Solution:
To get the sum of weighted terms, multiply each average by the number of students that had that average and then sum them up.
80 × 10 + 60 × 15 = 800 + 900 = 1700
Total number of terms = Total number of students = 25
Using the formula:
Weighted Average= $\dfrac{Sum \;of \;Weighted \;Terms}{Total \; Number\; of \;Terms}$
=$\dfrac{1700}{25}$
=68
Exercise:
Weighted Average= $\dfrac{Sum \;of \;Weighted \;Terms}{Total \; Number\; of \;Terms}$
Weighted Average= $\dfrac{3×90+2×80}{5}$
Weighted Average= $\dfrac{430}{5}$
=86
Average Speed:
Average Speed = $\left(\dfrac{Total\; Distance}{Total\;Time}\right)$
Suppose a man covers a certain distance at x kmph and an equal distance at y kmph
.Then, the average speed during the whole journey is = $\left(\dfrac{2xy}{x+y}\right)$
Example 1:
In travelling from city A to city B, John drove for 1 hour at 50 mph and for 3 hours at 60 mph. What was his average speed for the whole trip?
Solution:
The total distance is 1×50+3×60=230. And the total time is 4 hours.
Average Speed = $\left(\dfrac{Total\; Distance}{Total\;Time}\right)$
=$\dfrac{230}{4}$
=57.5
Example 2:
Distance between two stations A and B is 778 km. A train covers the journey from A to B at 84 km per hour and returns back to A with a uniform speed of 56km per hour. Find the average speed of the train during the whole journey?
Solution:
If a car covers a certain distance at x kmph and an equal distance at y kmph. Then,
Average speed of the whole journey =$\left(\dfrac{2xy}{x+y}\right)kmph.$
=$\dfrac{2×84×56}{84+56}$
=$\dfrac{2×84×56}{140}$
=$\dfrac{2×21×56}{35}$
=$\dfrac{2×3×56}{5}$
=$\dfrac{336}{5}$
=67.2
Exercise:
Average Speed = $\left(\dfrac{Total\; Distance}{Total\;Time}\right)$
$\dfrac{50 × 3 + 60 × 2}{5}$
$\dfrac{270}{5}$
= 54
The average speed is 54 miles per hour.