Analysis of Central Tendencies and Price Indices (Laspeyres & Paasche)

This report presents an analysis of descriptive statistics, including mean, median, standard deviation, and mode, applied to data from five companies trading in wine and cheese. The study utilizes data from September and October 2008, with September as the base period. The report calculates and compares Laspeyres and Paasche price indices, demonstrating how these indices measure price changes over time. The Laspeyres index is calculated by comparing the cost of the base period basket in the current period, while the Paasche index uses current-period weighting. The report finds that the Laspeyres index records a greater change in product costs than the Paasche index because it does not account for substitution away from products with higher price increases. Results include the Laspeyres and Paasche indices for each company, along with a discussion of the findings and relevant references.

Central Tendencies and Index numbers

Table of Contents
Introduction.................................................................................................................................................3
Data.............................................................................................................................................................5
Data analysis................................................................................................................................................7
Laspeyres Index.....................................................................................................................................10
Paasche Index........................................................................................................................................12
Discussion..................................................................................................................................................14
References.................................................................................................................................................16

Introduction
This paper provides a report on descriptive statistics, Laspeyres index and Paasche’s index on the
goods consumed from five companies.
Descriptive statistics are used to describe basic features such as mean, median, standard
deviation and mode of the data under study (Mary L. McHugh, 2009). For our study we shall use
descriptive statistics to provide the mean and median for our data under study.
The CPI is calculated for a fixed market basket (Liang, 2013). It measures the change in the cost
of the market basket from the base year until the current year (M.W.L. Chan, 2008). Two indices
were used for the purposes of this study; the Laspeyres and Paasches price indices.
The Laspeyres price index is used in business statistics for measuring the price development of
goods and services consumed in the base period (Slesnick, 1991). It tries to find out how much a
quantity of goods and services consumed during the base period would cost in the current period
(Siegel, 1941). The change in the price of quantities consumed basing on the base year provide
the Laspyeres price index. It is calculated as shown below;
PL= ∑ ( pc , tn )∗( qc , t 0 )
∑ ( pc , t 0 )∗( qc , t 0 )
Where PL is the Laspyres price index, pc,tn the price of the commodity at current calculation
time, qc,t0 is the quantity of the product consumed at base time and pc,t0 is the price of the
commodity at base time.

Similar to the Laspeyres price index, the Paasches price index measures current price of quantity
of commodities consumed relative to those of a selected base period (Hill, 2006). However,
Paasches price index uses current-period weighting (KARMEL, 1945).

Data
Quantity and price data were collected from five companies, A, B, C. D and E trading in the
commodities wine and cheese for the months of September and October 2008. The companies
were randomly drawn from a wide market of companies that deal in drinks.
Company Year
wine cheese
Quantity Price per
bottle
Quantity Price per
wheel
Company A September, 2008 50 bottles $0.50 100 wheels $1.00
October, 2008 45 bottles $2.00 150 wheels $1.20
Company B September, 2008 55 bottles $0.90 110 wheels $1.10
October, 2008 50 bottles $0.80 120 wheels $0.90
Company C September, 2008 52 bottles $1.00 105 wheels $0.95
October, 2008 57 bottles $1.10 100 wheels $1.20
Company D September, 2008 49 bottles $1.20 110 wheels $1.20
October, 2008 52 bottles $1.20 115 wheels $1.15
Company E September, 2008 45 bottles $0.90 140 wheels $1.30
October, 2008 46 bottles $1.20 150 wheels $1.30
The month September of 2008 was taken as the base year for our analyses, the market basket
contained only two items, wine and cheese, and the quantities consumed in September, 2008 and
October, 2008 for the five companies are as shown in the table below;

The companies’ prices for wine and cheese for the months September and October, 2008 are; For
company A, the price of cheese increased from $1.00 per wheel of cheese in September to $1.20
per wheel in October and the price of wine increased from $0.50 per bottle to $2.00 per bottle.
For company B, the price of cheese decreased from $1.10 per wheel of cheese in September to
$0.90 per wheel in October and the price of wine decreased from $0.90 per bottle to $0.80 per
bottle.
For company C, the price of cheese increased from $0.95 per wheel of cheese in September to
$1.20 per wheel in October and the price of wine increased from $1.00 per bottle to $1.10 per
bottle.
For company D, the price of cheese decreased from $1.20 per wheel of cheese in September to
$1.15 per wheel in October and the price of wine remained at $1.20 per bottle for September and
October.
For company E, the price of cheese remained at $1.30 per wheel of cheese in September and
October, 2008 and the price of wine increased from $0.90 in September, 2008 per bottle to $1.20
per bottle in October, 2008.

Data analysis
The mean quantity of wine consumed for the months September and October, 2008 for the five
companies is 50.1. The median of the quantity of wine consumed is 50.
Mean 50.1
Standard Error 1.286252092
Median 50
Mode 50
Standard Deviation 4.067486256
Sample Variance 16.54444444
Kurtosis -0.746085319
Skewness 0.256584385
Range 12
Minimum 45
Maximum 57
Sum 501
Count 10
The average price of wine for the five companies over the two months is $1.08 and the median
price for wine is 1.05.
Mean 1.08
Standard Error 0.123648247

Median 1.05
Mode 1.2
Standard Deviation 0.391010088
Sample Variance 0.152888889
Kurtosis 3.489535548
Skewness 1.288590141
Range 1.5
Minimum 0.5
Maximum 2
Sum 10.8
Count 10
The average quantity of cheese consumed for the five companies per month is 120. The median
of cheese quantity consumed is 112.5.
Mean 120
Standard Error 6.191391874
Median 112.5
Mode 100
Standard Deviation 19.57890021
Sample Variance 383.3333333
Kurtosis -1.11436673
Skewness 0.749475859

Range 50
Minimum 100
Maximum 150
Sum 1200
Count 10
The man price of cheese for the five companies is $1.13 and the median is 1.175.
Mean 1.13
Standard Error 0.044221664
Median 1.175
Mode 1.2
Standard Deviation 0.13984118
Sample Variance 0.019555556
Kurtosis -0.985059862
Skewness -0.475376944
Range 0.4
Minimum 0.9
Maximum 1.3
Sum 11.3
Count 10
Laspeyres Index
Company A

The Laspeyres price index for October, 2008 is given by 100*{(cost of base time basket in
October) / (cost of base time basket in September)}
Laspeyres Oct =100∗( 100∗$ 1.20+50∗$ 2
100∗$ 1.00+50∗$ 0.50 )=176
The rate of change due to the Laspeyres price is;
176−100
100 =0.76∨76 % .
Company B
The Laspeyres price index for October, 2008 is given by;
Laspeyres Oct =100∗(110∗$ 0.90+55∗$ 0.80
110∗$ 1.10+55∗$ 0.90 )=83.87
The rate of change due to the Laspeyres price is;
83.87−100
100 =−0.1613∨−16.13 % .
Company C
The Laspeyres price index for October, 2008 is given by;
Laspeyres Oct =100∗(105∗$ 1.20+52∗$ 1.10
105∗$ 0.95+52∗$ 1.10 )=120.72
The rate of change due to the Laspeyres price is;
120.72−100
100 =0.2072∨20.72 % .
Company D

The Laspeyres price index for October, 2008 is given by;
Laspeyres Oct =100∗(110∗$ 1.15+49∗$ 1.20
110∗$ 1.20+49∗$ 1.20 )=97.12
The rate of change due to the Laspeyres price is;
97.12−100
100 =−0.0288∨−2.88 % .
Company E
The Laspeyres price index for October, 2008 is given by;
Laspeyres Oct =100∗( 140∗$ 1.30+45∗$ 1.20
140∗$ 1.10+45∗$ 0.90 ) =106.07
The rate of change due to the Laspeyres price is;
106.07−100
100 =0.0607∨6.07 % .
Paasche Index
Company A
The Paasche price index for October, 2008 is given by;

Laspeyres Oct =100∗( 150∗$ 1.20+45∗$ 2.00
150∗$ 1.00+45∗$ 0.50 ) =156.52
The rate of change due to the Paasche price is;
156.52−100
100 =0.5652∨56.52 % .
Company B
The Paasche price index for October, 2008 is given by;
Laspeyres Oct =100∗(120∗$ 0.90+50∗$ 0.80
120∗$ 1.10+50∗$ 0.90 )=83.62
The rate of change due to the Paasche price is;
83.62−100
100 =−0.1638∨−16.38 % .
Company C
The Paasche price index for October, 2008 is given by;
Laspeyres Oct =100∗( 100∗$ 1.20+57∗$ 1.10
100∗$ 0.95+57∗$ 1.00 )=120.20
The rate of change due to the Paasche price is;
179.62−100
100 =0.2020∨20.20 % .
Company D
The Paasche price index for October, 2008 is given by;