Indices
Indices are statistical measures used to represent the relative change in a set of data points. They provide ways to simplify complex data by summarizing it into a single figure, making comparisons and trends easier to understand. They are commonly used in mathematics and finance to represent changes in values, quantities, or performance over time.
Types of Indices
- Mathematical Indices: These include exponents or powers, where a number is multiplied by itself multiple times. For instance, \(2^3 = 2 \times 2 \times 2 = 8\)
- Stock Market Indices: These track the performance of a group of stocks in the market
- Economic Indices: Used in economics to measure price changes, production levels, or income trends
Time-Series Graphs
A Time-Series Graph is a chart used to show how data changes over time. It helps identify trends, patterns, and fluctuations.
Characteristics of Time-Series Graphs
- X-Axis (Horizontal): Represents time (days, months, years)
- Y-Axis (Vertical): Represents the measured data (i.e. temperature, stock prices)
- Line Graph: The most common type of time-series graph, where points are connected by lines to show trends
Uses of Time-Series Graphs
- Weather Forecasting: Shows temperature changes over time.
- Stock Market Analysis: Tracks stock price movements
- Economics: Measures inflation, employment rates, or sales trends
- Scientific Studies: Observes growth patterns in biology or chemical reactions
Consumer Price Index (CPI)
The Consumer Price Index (CPI) measures the average change in prices of goods and services over time. It is used to track inflation and helps in adjusting wages, pensions, and government policies.
Inflation occurs when the price of items like food, clothes, and gas increase over time. This means that a particular amount of money no longer carries the same value. If inflation happens, you might need more money to buy the same things you bought before.
A fixed "basket" of goods and services (like food, transport, healthcare) is chosen. The prices of these items are collected regularly and compared to a base year. The CPI can be determined algebraically as such:How CPI is Calculated?
Example
A market basket contains 3 loaves of bread, 2 gallons of gas, and 4 dozen eggs. The prices for these items in the base year and the current year are as follows:
Base Year Prices:
- Bread: $2 per loaf
- Gas: $3 per gallon
- Eggs: $2.50 per dozen
Current Year Prices:
- Bread: $2.80 per loaf
- Gas: $4 per gallon
- Eggs: $3 per dozen
Determine the inflation rate of the current year relative to the base year.
First, we can calculate the cost of the basket in the base year. We can do so by first determining the costs of the individual items:
\(\text{Bread: } 3 \times $2 = $6 \)
\(\text{Gas: } 2 \times $3 = $6\)
\(\text{Eggs: } 4 \times $2.50 = $10\)
We can then determine the total cost of the basket in the base year:
\(\text{Total}_{\text{Base}}: $6 + $6 + $10 = $22\)
Next, we can calculate the cost of the basket in the current year. We can do so by first determining the costs of the individual items:
\(\text{Bread: } 3 \times $2.80 = $8.40 \)
\(\text{Gas: } 2 \times $4 = $8\)
\(\text{Eggs: } 4 \times $3 = $12\)
We can then determine the total cost of the basket in the current year:
\(\text{Total}_{\text{Current}}: $8.40 + $8 + $12 = $28.40\)
Then, we can calculate the CPI:
\(CPI = \left( \cfrac{\text{Current Cost of Basket}}{\text{Base Year Cost of Basket}} \right) \times 100\)
\(= \left( \cfrac{$28.40}{$22} \right) \times 100\)
\(= 129.09 \)
Finally, we can calculate the percentage change in prices:
\(\text{Inflation Rate} = \left( \cfrac{129.09 - 100}{100} \right) \times 100 \)
\(= 29.09\% \)
Therefore, we can determine a CPI of 129.09 means that prices have increased by 29.09% compared to the base year.
Outlined below are some of the main uses of CPI:
- Helps governments adjust salaries for inflation
- Affects interest rates and economic policies
- Shows if the cost of living is increasing or decreasing
A basket contains 2 loaves of bread and 1 gallon of gas. The prices for these items in the base year and the current year are as follows:
Base Year Prices:
- Bread: $2 per loaf
- Gas: $3 per gallon
Current Year Prices:
- Bread: $2.50 per loaf
- Gas: $3.50 per gallon
What is the Consumer Price Index (CPI) for the current year, and what does it indicate about price changes compared to the base year?
First, we can calculate the cost of goods in the base year. We can do so by first determining the costs of the individual items:
\(\text{Bread}: 2 \times $2 = $4\)
\(\text{Gas}: 1 \times $3 = $3\)
We can then determine the total cost of goods in the base year:
\(\text{Total}_{\text{Base}}: $4 + $3 = $7\)
Next, we can calculate the total cost of the goods in the current year. We can do so by first determining the costs of the individual items:
\(\text{Bread}: 2 \times $2.50 = $5\)
\(\text{Gas}: 1 \times $3.50 = $3.50\)
We can then determine the total cost of goods in the current year:
\(\text{Total}_{\text{Current}}: $5 + $3.50 = $8.50\)
Then, we can calculate the CPI:
\(CPI = \left( \cfrac{\text{Current Cost of Basket}}{\text{Base Year Cost of Basket}} \right) \times 100\)
\(= \left( \cfrac{$8.50}{$7} \right) \times 100\)
\(= 121.43\)
Therefore, we can determine the CPI for the current year is 121.43. This means prices have increased by about 21.43% compared to the base year.