Clear-Sighted Statistics: Module 6: Index Numbers (slides)

City University of New York (CUNY) City University of New York (CUNY)

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Open Educational Resources Queensborough Community College

2020

Clear-Sighted Statistics: Module 6: Index Numbers (slides) Clear-Sighted Statistics: Module 6: Index Numbers (slides)

Edward Volchok CUNY Queensborough Community College

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Clear-Sighted Statistics

Index NumbersModule 6

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Clear-Sighted Statistics

Lecture Objectives

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Understand why we use

index numbers

Know how to interpret index

numbers

Calculate simple

unweighted index numbers

Use Index numbers to

calculate Coefficient of

Variation

Use index numbers to calculate a geometric

mean

Construct weighted

indices

Identify special-purpose

indices: CPI, PPI, DJIA,

S&P500, etc.

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Clear-Sighted Statistics

Interpreting Index Numbers & Simple Index Numbers

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Clear-Sighted Statistics

Index numbers invented in 1764

Inventor: Italian economist Giovanni Rinaldo (Count of Carli)

Used index numbers to compare the prices of grain, wine, and oil for a 250-year period

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Count Giovanni Rinaldo1720 - 1795

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Clear-Sighted Statistics

Indices are widely used

Seen in business, finance, economics, politics, and other social sciences

Used to compare relative differences among quantitative variables

Used when calculating the coefficient of variation

Used when calculating the geometric mean when data contains negative numbers

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Grammar lesson

The plural of index is either indices or indexes

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Simple Index Numbers

Compare the relative difference between two numbers

One value is considered the “base”

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Formula for Simple Index Numbers

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IndexNumber= SelectedValueBaseValue

x100

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Clear-Sighted Statistics

Formula for Simple Index Numbers

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P=Pt

Pox100

Where: P = IndexPt = Selected ValuePo = Base Value

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Clear-Sighted Statistics

Example: CityMetric – World’s 10 largest subway systems, number of stations

Indices based on the 10-system average

Index of ≈100 is average

Index ≤85 is below average

Index ≥115 is above average

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Clear-Sighted Statistics

The Indices

10Number of Subway stations for 10 international cities

*Includes lines 1 - 9

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More on interpreting index numbers

An index of 200 is double the base

An index of 300 is triple the base

An index of 75 is 75% of the base

An index of 50 is half of the base

An index of 25 is a quarter of the

base

The smallest possible index

is zero

Nice, France would index at 0 because it has no subway system

There are no negative index

numbers

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Clear-Sighted Statistics

Coefficient of VariationReported as an index number

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Coefficient of Variation

Used for comparing dispersion of two of more ratio level variables that have different scales

Can be reported as a percentage, decimal, or an index number

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Coefficient of Variation index formulas

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CV= σµ*100CV= s

X*100

Where: s = Sample Standard DeviaDonσ = PopulaDon Standard DeviaDonXG = Sample Meanμ = PopulaDon Mean

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Clear-Sighted Statistics

Coefficient of Variation dataPrice of a McDonald’s Big Mac in 20 countries

Monthly mobile data usage in gigabytes in the same 20 countries

Mobile data usage is 115% more variable than the price of a Big Mac, found by (73.78/34.29)*100

Big Mac data is only 46% as variable as Mobile Data usage, found by (34.48/73.78)*100

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Big Mac

Mobile Data Formula

Sample Std. Dev. $1.29 2.4Sample Mean $3.78 3.2

CV 34.29 73.78 (SD/Mean)100

73.7834.29

*100= 215.15≈215

34.2973.78

*100= 46.48≈ 46

CV= sX*100

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Clear-Sighted Statistics

Index Numbers & the Geometric Mean

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Using indices to calculate the geometric mean

The geometric mean is the the nth root product of n numbers

The geometric mean is used in investment analysis to calculate the average rate of return

BIG PROBLEM: Can only be calculated with non-negative values

When investments lose money, we have negaRve returns and the geometric mean cannot be calculated

SOLUTION: Convert returns to indices

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Calculating the geometric mean

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Column B shows data formatted as percentages

Using Excel’s GEOMEAN function, returns the #NUM! error (invalid number format error)

Using indices (Column F), the geometric mean is 94.33, a loss of 5.67%

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Calculate Unweighted Price IndicesSimple Price Index and Aggregate Price Index

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Simple Price Index

Compare changes in price over two periods for a market basket of items

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P=Pi∑

n*100

Where: Σ indicates the operation of additionP is the average index Pi are the individual indicesn is the number of indices

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Clear-Sighted Statistics

Simple Price Index (Example)

Retail prices for 2009 and 2019 for four chocolate manufacturers with a fifth category for small brands called “All Others”

Simple indices are shown in Column D and the formulas are in Column E

Simple price index = 147, shown in cell D7, is the mean of the five indices

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P=Pi∑

n*100

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Clear-Sighted Statistics

Simple Aggregate Price IndexPrices, not indices, for each item are summed then

the index is calculated from the sum of the base and selected periods

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P=Pt∑Po∑

*100

Where: ΣPo is the sum of the values in the base periodΣPt is the sum of the values in the selected period

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Clear-Sighted Statistics

Simple Aggregate Price Index (Example)

Retail prices for 2009 and 2019 for four chocolate manufacturers with a fifth category for small brands called “All Others”

Indices are based on the sum of the prices in the selected period in the numerator (2019) and the sum of the base period in the denominator

Simple aggregate price index = 146, shown in cell D7

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P=Pt∑Po∑

*100

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Weighted IndicesLespeyres, Paasche, Fisher Ideal, and Value Indices

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Why use a weighted index?

Using a weighted price index is often more appropriate than an unweighted index

Each variable in the index is adjusted to account for the quality

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Laspeyres Index (Etienne Laspeyres)

This is a base period quantity index because its weights use the base period’s quantities and price

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Where: Pt is the Price for the Observed PeriodPo is the Price for the Base PeriodQt is the Quantity for the Observed PeriodQo is the Quantity for the Base Period

PL =PtQ o∑PoQ o∑

*100

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Clear-Sighted Statistics

Laspeyres Index (Example)

Price for the market basket rose ≈42% (index: 141.52)

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PP =PtQ t∑PoQ t∑

*100

Paasche Index (Hermann Paasche)

This is a current period weighted index because it uses the current or observed period weightings

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Where: Pt is the Price for the Observed PeriodPo is the Price for the Base PeriodQt is the Quantity for the Observed PeriodQo is the Quantity for the Base Period

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Clear-Sighted Statistics

Paasche Index (Example)

Price for the market basket rose ≈37% (index: 137.34)

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Laspeyres Vs. Paasche

Same market basket, different indices

Laspeyres = 142 Paasche = 137

Laspeyres tends to over-estimate price

Paasche tends to under-estimate price

Laspeyres is used more often based on cost and ease of calculation

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Clear-Sighted Sta0s0cs

Fisher’s Ideal Index (Irving Fisher)

Overcomes issue with Laspeyres and Paasche indices

Geometric mean of the product of the Laspeyres and Paasche indices

With 2 periods: Square root of the product of this indices

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PF= Laspeyres( )* Paasche( )n

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Clear-Sighted Statistics

Fisher’s Ideal Index (Example)

Fisher’s Index: 139.41

Laspeyres: 141.52

Paasche: 137.34

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Value IndexLaspeyres and Paasche measure changes in weighted price

Value Index weights both price and quantity

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Where: V = Value IndexΣPtQt is the sum of the Prices for the selected period times the Quantities for the selected periodΣPoQo the sum of the Prices for the base period times the Quantities for the base period

V=PtQ t∑PoQ o∑

*100

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Clear-Sighted Statistics

Value Index (Example)

Price of the market basket rose ≈47%

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Special Purpose IndicesPublished by governments and financial institutions

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Consumer Price Index (CPI)

Published by the US Dept. of Labor’s Bureau of Labor Statistics

“…measure of the average change over time in the prices paid by urban consumers for a market basket of consumer goods and services. Indexes are available for the U.S. and various geographic areas. Average price data for select utility, automotive fuel, and food items are also available.”

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Producer Price Index (PPI)

Published by the US Dept. of Labor’s Bureau of Labor Statistics

“…measures the average change over time in the selling prices received by domestic producers for their output. The prices included in the PPI are from the first commercial transaction for many products and some services.”

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Dow Jones Industrial Average (DJIA)

Published by S&P Dow Jones LLC Started in 1896

Measures daily stock prices of 30 large companies on the New York Stock Exchange and Nasdaq

Proxy for health of financial markets and US economy

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Clear-Sighted Statistics

S&P 500

Barometer for large capitalization American equities

Top 500 companies based on market capitalization

Covers approximately 80% of available market capitalizaaon

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Russell 2000

Published by FTSE Russell, a subsidiary of the London Stock Exchange Group

Stock market index composed of 2,000 publicly-traded small-capitalization American firms

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NASDAQ-100 IndexPublished by the National Association of Securities Dealers

NASDAQ is an electronic stock exchange

Includes 100 of the largest non-financial companies listed on Nasdaq based on market capitalization

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NIKKEI 225

Stock market index for the Tokyo Stock Exchange

Price-weighted index that measures the performance of 225 publicly-traded companies for a broad selection of industrial sectors

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Summary

Index numbers measure the relative difference between a base value and a selected value

There are many formulas for calculating index numbers

The correct formula depends on the analyst’s objectives and the nature of the data

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Except where otherwise noted Clear-Sighted Statistics is licensed under

a Creative Commons License. You are free to share derivatives of this work

for non-commercial purposes only. Please attribute this work to Edward Volchok.

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