UNIT 5 | unit 5 averages

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UNIT 5

AVERAGES

5.1 Simple, weighted and statistical averages, arithmetic mean, harmonic mean, geometric mean.

Mean

The mean is the arithmetic average, also called as arithmetic mean.
Mean is very simple to calculate and is most commonly used measure of the center of data.
Means is calculated by adding up all the values and divided by the number of observation.

Computation of sample mean -

If X1, X2, ………………Xn are data values then arithmetic mean is given by

Computation of the mean for ungrouped data

Example 1 – The marks obtained in 10 class test are 25, 10, 15, 30, 35

The mean = X = 25+10+15+30+35 = 115 =23

5 5

Analysis – The average performance of 5 students is 23. The implication is that students who got below 23 did not perform well. The students who got above 23 performed well in exam.

Example 2 – Find the mean

Xi	9	10	11	12	13	14	15
Freq (Fi)	2	5	12	17	14	6	3

Xi	Freq (Fi)	XiFi
9	2	18
10	5	50
11	12	132
12	17	204
13	14	182
14	6	84
15	3	45
	Fi = 59	XiFi= 715

Then, N = ∑ fi = 59, and ∑fi Xi=715

X = 715/59 = 12.11

Mean for grouped data/ Weighted Arithmetic Mean

Grouped data are the data that are arranged in a frequency distribution

Frequency distribution is the arrangement of scores according to category of classes including the frequency.

Frequency is the number of observations falling in a category

The formula in solving the mean for grouped data is called midpoint method. The formula is

Where, X = Mean

Xm = midpoint of each class or category

f = frequency in each class or category

∑f Xm = summation of the product of fXm

Example 3 – the following data represent the income distribution of 100 families. Calculate mean income of 100 families?

Income	30-40	40-50	50-60	60-70	70-80	80-90	90-100
No. Of families	8	12	25	22	16	11	6

Solution:

Income	No. Of families	Xm (Mid point)	FXm
30-40	8	35	280
40-50	12	34	408
50-60	25	55	1375
60-70	22	65	1430
70-80	16	75	1200
80-90	11	85	935
90-100	6	95	570
	n = 100		∑f Xm = 6198

X = ∑f Xm/n = 6330/100 = 63.30

Mean = 63.30

Example 4 – Calculate the mean number of hours per week spent by each student in texting message.

Time per week	0 - 5	5 - 10	10 - 15	15 - 20	20 - 25	25 – 30
No. Of students	8	11	15	12	9	5

Solution:

Time per week (X)	No. Of students (F)	Mid point X	XF
0 - 5	8	2.5	20
5 – 10	11	7.5	82.5
10 - 15	15	12.5	187.5
15 - 20	12	17.5	210
20 - 25	9	22.5	202.5
25 – 30	5	27.5	137.5
	60		840

Mean = 840/60 = 14

Example 5 –

The following table of grouped data represents the weights (in pounds) of all 100 babies born at a local hospital last year.

Weight (pounds)	Number of Babies
[3−5)	8
[5−7)	25
[7−9)	45
[9−11)	18
[11−13)	4

Solution:

Weight (pounds)	Number of Babies	Mid point X	XF
[3−5)	8	4	32
[5−7)	25	6	150
[7−9)	45	8	360
[9−11)	18	10	180
[11−13)	4	12	48
	100		770

Mean = 770/100 = 7.7

Geometric mean

Geometric mean is a type of mean or average, which indicates the central tendency of a set of numbers by using the product of their values.

Definition

The Geometric Mean (G.M) of a series containing n observations is the nth root of the product of the values.

For ungrouped data

Geometric Mean, GM = Antilog ∑logxi

Example 1 – find the G.M of the values

X	Log X
45	1.653
60	1.778
48	1.681
65	1.813
Total	6.925

GM = Antilog ∑logxi

= Antilog 6.925/4

= Antilog 1.73

= 53.82

For grouped data

Geometric Mean, GM = Antilog ∑ f logxi

Example 2 – calculate the geometric mean

X	f
60 – 80	22
80 – 100	38
100 – 120	45
120 – 140	35

Solution

X	f	Mid X	Log X	f log X
60 – 80	22	70	1.845	40.59
80 – 100	38	90	1.954	74.25
100 – 120	45	110	2.041	91.85
120 – 140	35	130	2.114	73.99
Total	140			280.68

GM = Antilog ∑ f logxi

= antilog 280.68/140

= antilog 2.00

GM = 100

Example 3 – calculate geometric mean

Class	Frequency
2-4	3
4-6	4
6-8	2
8-10	1

Solution

Class	Frequency	x	Log x	Flogx
2-4	3	3	1.0986	3.2958
4-6	4	5	1.2875	6.4378
6-8	2	7	0.5559	3.8918
8-10	1	9	0.2441	2.1972
	10			15.8226

GM = Antilog ∑ f logxi

= antilog 15.8226/10

= antilog 1.5823

GM = 4.866

Harmonic mean

Harmonic mean is quotient of “number of the given values” and “sum of the reciprocals of the given values

For ungrouped data

Example 1 - Calculate the harmonic mean of the numbers 13.2, 14.2, 14.8, 15.2 and 16.1

Solution

X	1/X
13.2	0.0758
14.2	0.0704
14.8	0.0676
15.2	0.0658
16.1	0.0621
Total	0.3147

H.M of X = 5/0.3147 = 15.88

Example 2 - Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5} ?

X	1/X
8	0.125
9	0.111
6	0.167
11	0.091
10	0.100
5	0.200
Total	0.794

H.M of X = 6/0.794 = 7.560

For grouped data

Example 3 - Calculate the harmonic mean for the below data

Marks	30-39	40-49	50-59	60-69	70-79	80-89	90-99
F	2	3	11	20	32	25	7

Solution

Marks	X	F	F/X
30-39	34.5	2	0.0580
40-49	44.5	3	0.0674
50-59	54.4	11	0.2018
60-69	64.5	20	0.3101
70-79	74.5	32	0.4295
80-89	84.5	25	0.2959
90-99	94.5	7	0.0741
Total		100	1.4368

HM = 100/1.4368 = 69.59

Example 4 – find the harmonic mean of the given class

Ages	4	5	6	7
No. Of students	6	4	10	9

Solution

X	F	f/x
4	6	1.50
5	4	0.80
6	10	1.67
7	9	1.29
	29.00	5.25

HM = 29/5.25 = 5.5

Example 5 – calculate harmonic mean

Class	Frequency
2-4	3
4-6	4
6-8	2
8-10	1

Solution

Class	Frequency	x	f/x
2-4	3	3	1
4-6	4	5	0.8
6-8	2	7	0.28
8-10	1	9	0.11
	10		2.19

Harmonic mean = 10/2.19 = 4.55

Merits of mean

It is rigidly defined
It is easy to understand and easy to calculate
It is based upon all values of the given data
It is capable of future mathematical treatment
It is not much affected by sampling fluctuation

Demerits of mean

It cannot be calculated if any observation are missing
It cannot be calculated for open end classes
It is effected by extreme values
It cannot be located graphically
It may be number which is not present in the data

5.2 Profit and loss

Terms and formulae, problem involving cost price, selling price, trade discount, cash discount

Cost Price (CP) - The price at which an article is purchased.

Selling Price (SP) - This is the price at which an article is sold.

Profit - If the selling price is more than the cost price, then there will be profit

Profit = Selling price (SP) – Cost price (CP)

Profit % = (profit100)/CP

Loss - If the selling price is less than the cost price, then there will be loss

Loss = Cost price (CP) – Selling price (SP)

Loss% = (Loss 100)/CP

Marked Price - This is the price marked as the selling price on an article, also known as the listed price

Note- Profit or Loss is always calculated on the cost price.

Discount: This is the reduction in price offered on the marked or listed price.

Example: Suppose the profit percentage is 80% of the cost. If the cost further increases by 20% but the selling price remains the same, how much is the decrease in profit percentage?

Sol.

Let us suppose

CP = Rs. 100.

Then Profit = Rs. 80 and selling price = Rs. 180.

The cost increases by 20% → New CP = Rs. 120, SP = Rs. 180.

Profit % = 60/120 100 = 50%.

Therefore, Profit decreases by 30%.

Example: A dishonest shopkeeper sells his grocery using weights 15% less than the true weights and makes a profit of 20%. Find his total gain percentage.

Sol.

Suppose 1 kg of grocery bag.

Its actual weight is 85% of 1000 gm = 850 gm.

Let the cost price of each gram be Re. 1. Then the CP of each bag = Rs. 850.

SP of 1 kg of bag = 120% of the true CP

Therefore, SP = 120/100 1000 = Rs. 1200

Gain = 1200 – 850 = 350

Hence Gain % = 350/850 100 = 41.17%

Example: A shopkeeper offers a discount of 20% on the selling price. On a special sale day, he offers an extra 25% off coupon after the first discount. If the article was sold for Rs. 3600, find the marked price of the article

Sol.

Let the marked price of the article be x.

First a 20% discount was offered, on which another 25% discount was offered.

So, 75% of 80% of x = 3600

75/100 80/100 x = 3600 → x = 6000.

So the article was marked at Rs. 6000.

Example: The cost price of 30 articles is equal to the selling price of 40 articles. What is the profit or loss percentage?

Sol.

CP of 30 articles = SP of 40 articles Or, CP/SP = 30/40 =3/4

Or, 1 – CP/SP = 1- 3/4 = 1/4

So, Loss percentage

= (l – CP/SP) x 100 = 1/4x 100 = 25%

Trade discount-

Trade discount is a certain percentage a manufacturer is willing to reduce its list price for wholesalers or retailers.

Trade discounts are given to wholesalers that order large quantities of a product as well as retailers with good relationships with the manufacturer. Purchase discounts or Cash discount are based on payment plans not order quantities.

Trade discounts are offered on bulk purchases by traders, wholesalers, distributors or retailers and not to the end consumers.

Trade discounts are generally offered at varied rates depending on the volume of sale

And

Example: Sweety purchases a set of toys that lists for Rs. 950 and it has a trade discount of 30%, then find the net price.

Sol.

As we know that-

Then

Net price = 950 – 285 = 665

Example: If Raheem will buy a table that lists for Rs. 2000 and it has a trade discount of 50%. Then how much will he pay?

Sol.

As we know that-

Then

Net price = 2000 – 1000 = 1000

Cash discountis allowed to stimulate instant payment of the goods purchased.

Cash Discount is referred to as a discount, allowed to customers by the seller at the time of making the payment of purchases, as a reduction in the invoice price of the commodity.

Both the buyers and sellers keep a proper record of such discount in their books of accounts.

Sellers offer cash discounts to their buyers as an incentive to encourage early payment.

Ordinary dating method-

A credit term of [2/10, n/30] means that you will get a discount of 2% if you clear your account within 10 days

Example: A person received an invoice for Rs. 3,000 dated 22 November 2019 with terms [2/10, n/30]. He paid the whole amount on 30 November 2019. How much did he effectively paid for the bill?

Sol.

Date of Invoice: 22nd November 2019
Day 1 of the cash discount period. : 23rd November 2019
Last day of the Cash discount period: 31st November 2019
Date of payment: 30th November 2019

Cash discount.= Price Discount rate

= Rs. 3,000 2/100

= Rs. 60

Amount effectively paid by the person = Bill value - Cash discount

= 3,000 - 60

= Rs. 2,940

Key takeaways-

Cost Price (CP) - The price at which an article is purchased
Selling Price (SP) - This is the price at which an article is sold
Profit = Selling price (SP) – Cost price (CP)
Profit = Selling price (SP) – Cost price (CP)
Loss = Cost price (CP) – Selling price (SP)
Loss% = (Loss 100)/CP
Marked Price - This is the price marked as the selling price on an article, also known as the listed price
Discount: This is the reduction in price offered on the marked or listed price.
Trade discount is a certain percentage a manufacturer is willing to reduce its list price for wholesalers or retailers.
Trade discounts are generally offered at varied rates depending on the volume of sale

Introduction to commission and brokerage- problems on commission and brokerage

The amount received by the agency for providing its services is known as ‘brokerage’ or ‘commission’.

For example- placement agencies, marriage bureaus etc.

An individual can also run this type of business

Note- An individual who acts as a ‘middle man’ between a seller and buyer is known as broker.

Brokerage is paid by purchaser, Commission is paid by seller. Therefore

Net amount paid by purchaser = Sale price + brokerage

Net amount received by seller = Sale price - commission

Commission % = Commission100 /Selling price

Selling Price = Commission100/ Commission %

Commission (Brokerage) = Commission%Selling price/100

Example: The Price of a book is Rs. 15.75. A book seller sells 1200 books and pays Rs 17,860.50 to the book publisher after deducting his commission.

Find the commission rate.

Sol.

Here we have total commission,

We have to find the total sale price of books.

The total sale price of books = Number of Books Price = 120015.75 =18900

Commission = Sale price – Amount paid to the Publisher = 18900-17860.5 = 1039.5

Commission % = Commission100 /Selling price = 1039.5100/18900 = 5.5%

Example: Mahesh sold his car for Rs.68,000 with the help of an agent. If the commission paid by Mahesh is Rs 2550. Find rate of commission, and the net amount received by the Mahesh.

Sol.

Commission % = Commission100 /Selling price = 2550100/68000 = 3.75%

Net amount received by farmer = Sale price – Commission = 68000-2550 = Rs65, 450

Example: An agency pays 15% commission to a notebook distributor. The price of each notebook is Rs 3. If he sells 50 copies of notebook every day, find how much commission he receives in a month and also net amount received by the Agency?

Sol.

Price of each news paper = Rs 3.

Sale price for a day = Number copies sold in day price of Cost of news paper = 503 = Rs 150