Quantitative methods: basic statistics

Paasche’s price index number is calculated as:

$P_{01}=\frac{\sum p_1q_1}{\sum p_0q_1}\times 100$

where $p_0$ and $q_0$ are base year (2078) price and quantity, and $p_1$ and $q_1$ are current year (2080) price and quantity. Paasche’s method uses current year quantities as weights.

Now,

$P_{01}=\frac{311}{200}\times 100 = 155.5$

Interpretation: The Paasche price index number for 2080 with base year 2078 is 155.5. This means the general price level in 2080 is 55.5% higher than in 2078, when the quantities consumed in 2080 are used as weights. In other words, the cost of the current year’s basket of goods has increased by 55.5% compared with the base year.

It requires studying only a small representative part of the population instead of every unit, so it saves time, money and labour.

Statistics in a singular sense means the science or method of collecting, organising, presenting, analysing, and interpreting numerical data.

Primary data is the data that is collected firsthand by the investigator himself, directly from the field or respondents, for the first time for a specific purpose. It is original in nature and has not been previously collected or published by anyone else.

Range cannot be determined because the data set is not provided.

Coefficient of S.D. = $\dfrac{\text{S.D.}}{\text{Mean}} \times 100 = \dfrac{10}{400} \times 100 = 2.5\%$

Arithmetic mean

$\bar{x}=\frac{\sum fx}{N}=\frac{930}{40}=23.25$

Mean deviation about the mean

$MD_{\bar{x}}=\frac{\sum f\lvert x-\bar{x}\rvert}{N}=\frac{387.50}{40}=9.6875\approx 9.69$

Coefficient of mean deviation

$\frac{MD_{\bar{x}}}{\bar{x}}=\frac{9.6875}{23.25}=0.4167$

Mean deviation about mean $=9.69$ marks; coefficient of mean deviation $=0.4167$ (or $41.67\%$).

Laspeyres’ price index formula:

$P_{01}=\frac{\sum p_1q_0}{\sum p_0q_0}\times 100$

where $p_0$ = base year price (2019), $q_0$ = base year quantity (2019), and $p_1$ = current year price (2023).

Now,

$P_{01}=\frac{3225}{2825}\times 100 = 114.16 \text{ approximately}$

Result and analysis:

The Laspeyres’ price index number for 2023 taking 2019 as base year is 114.16.

Since the index is greater than 100, it shows that the general price level has increased by 14.16% in 2023 compared to 2019. In other words, a basket of goods that cost Rs 100 in 2019 costs about Rs 114.16 in 2023. The highest price rise is seen in commodity B, while the price of commodity D remains unchanged. Thus, there is a clear inflationary tendency in the economy during the period.

Primary data is preferred because it is collected first-hand by the researcher specifically for the present study, so it is original, more reliable, and directly suited to the research objective.

Coefficient of range $= \dfrac{L-S}{L+S} = \dfrac{25}{85} = 0.294$

Interpretation: The value is small (close to 0), so the dispersion between the largest and smallest values is low and the series is fairly consistent.

Let $A=35$ and class width $h=10$.

Mean:

Standard deviation:

Interpretation:
The average income of labour is about $35.67$ units and the standard deviation is about $12.50$ units. This means the incomes of labourers deviate from the mean by about $12.50$ units on average. The coefficient of variation is

Since this is fairly high, the income distribution among labour is not homogeneous; there is considerable variation/inequality in their incomes.

Statistics in plural sense refers to the numerical facts or data themselves — the quantitative information collected, such as figures on prices, production, income or population.

In the plural sense, "statistics" means the actual numerical statements of facts (data), while in the singular sense it means the science or method of collecting, organising, presenting, analysing and interpreting such data.

Using the empirical relation:

$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$

$25 = 3(23) - 2\bar{x}$

$25 = 69 - 2\bar{x}$

$2\bar{x} = 44 \Rightarrow \bar{x} = 22$

Arithmetic mean = 22

(a) Coefficient of variation (CV)

$CV=\frac{\text{Standard deviation}}{\text{Average wage rate}}\times 100\%$

For Company X:

$CV_X=\frac{13}{45}\times 100\%=28.89\%$

For Company Y:

$CV_Y=\frac{12}{40}\times 100\%=30\%$

(b) More consistent company

Company X has a lower coefficient of variation (28.89%) than Company Y (30%). A lower CV means less relative variability and greater uniformity in wage rates. Therefore, Company X’s wage rate is more consistent.

Two importance of statistics in economics:

1. Formulation of economic policies: Statistical data on income, employment, prices and production enable the government to frame sound fiscal and monetary policies.
2. Comparison of economic conditions: Statistics allows comparison of economic variables across regions or time periods, which is essential for economic analysis and planning.

Mean $=\dfrac{10+20+X+40+60}{5}=35$

$\dfrac{130+X}{5}=35$

$130+X=175$

$X=45$

X = Rs 45

Standard deviation (σ)

Let assumed mean $A = 55$.

Standard deviation = 6.04 marks

Why standard deviation is superior to mean deviation:

1. Algebraic treatment: Standard deviation can be used for further algebraic and statistical analysis, such as correlation, regression, and sampling. Mean deviation uses absolute values, so it cannot be treated algebraically.

2. Greater reliability: Standard deviation is based on all observations and gives more weight to extreme values because deviations are squared. Hence, it measures variability more accurately and is less affected by sampling fluctuations than mean deviation.

Let 2070 be the base year and 2078 be the current year.

Paasche’s price index number is:

where $p_0,q_0$ = base year price and quantity, and $p_1,q_1$ = current year price and quantity.

Now,

So, Paasche’s price index number is 136.07.

This means the price level in 2078 is about 36.07% higher than in 2070 when current year quantities are used as weights.

Difference between Paasche’s and Laspeyres’ price index numbers

1. Weights used:

   • Paasche’s index uses current year quantities $(q_1)$ as weights.
   • Laspeyres’ index uses base year quantities $(q_0)$ as weights.

2. Formulae:

3. Bias:

   • Paasche’s index generally has a downward bias, because it gives less weight to goods whose prices have risen more, as consumers substitute cheaper goods.
   • Laspeyres’ index generally has an upward bias, because it continues to give base year quantities to goods whose prices have risen.

4. Data requirement:

   • Paasche’s index requires current year quantity data.
   • Laspeyres’ index requires only base year quantity data.

For the given data, Laspeyres’ index would be:

Thus, Paasche’s index $(136.07)$ is lower than Laspeyres’ index $(147.85)$, which shows the usual downward bias of Paasche’s index.

Primary data is collected first-hand by the investigator through surveys, questionnaires or interviews, which requires trained enumerators, travel, printing and a lot of time, so it costs more than using data already collected by others.

Using the empirical relation among the averages:

$\text{Mode} = 3\,\text{Median} - 2\,\text{Mean}$

$25 = 3\,\text{Median} - 2(22) = 3\,\text{Median} - 44$

$3\,\text{Median} = 69 \Rightarrow \text{Median} = 23$

The median is 23.

(a) Coefficient of variation:

$CV = \frac{\sigma}{\bar{X}} \times 100$

Company X:

$CV_X = \frac{13}{40} \times 100 = 32.5\%$

Company Y:

$CV_Y = \frac{14}{50} \times 100 = 28\%$

(b) Company Y has more consistent wages, because its coefficient of variation (28%) is lower than that of X (32.5%). A smaller CV means less relative variation in wages around the average, even though Y has the larger standard deviation.

(1) The objectives of the study must be clearly defined before collecting primary data.
(2) The scope of the investigation — the area, population, and units to be studied — must be clearly decided in advance.

Two problems in using secondary data:

1. Reliability and accuracy: Secondary data was collected by someone else for another purpose, so its accuracy, method of collection and possible bias may be doubtful.
2. Relevance and uniformity: The units, definitions, coverage and time period may not match the present study's needs, and the data may be out of date.

Given: $Q_3 - Q_1 = 45,\qquad Q_3 + Q_1 = 75$

Solving, $Q_3 = \frac{75+45}{2}=60,\qquad Q_1 = \frac{75-45}{2}=15$

Coefficient of quartile deviation: $C.Q.D. = \frac{Q_3-Q_1}{Q_3+Q_1} = \frac{45}{75}=0.6$

In percentage, $0.6 \times 100 = 60\%$

Interpretation: The coefficient of quartile deviation is 0.6, i.e. 60%. This shows that the difference between the third and first quartiles is 60% of their sum, indicating a relatively high degree of variation or dispersion in the middle 50% of the data.

Given: Discrete series — wages ($X$) with frequencies ($f$).

Mean: $\bar X=\frac{\sum fX}{N}=\frac{1410}{24}=\text{Rs }58.75$

Standard deviation (actual mean method): $\sigma=\sqrt{\frac{\sum fd^2}{N}}=\sqrt{\frac{4062.5}{24}}=\sqrt{169.27}=\text{Rs }13.01$

Coefficient of variation: $CV=\frac{\sigma}{\bar X}\times 100=\frac{13.01}{58.75}\times 100=22.14\%$

Result: The mean wage of the 24 labourers is Rs 58.75 and the standard deviation is Rs 13.01.

Interpretation: On average, the wage of a labourer differs from the mean wage by about Rs 13. Since the coefficient of variation is only about 22%, the wages are fairly consistent — there is moderate, not extreme, inequality in the wage distribution.

Laspeyres's Price Index

Formula:

$P_{01}^{L} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100$

where $p_0$ = base year price, $p_1$ = current year price, $q_0$ = base year quantity (weights).

Data and computation:

Substitution:

$P_{01}^{L} = \frac{166}{102} \times 100 = 1.6275 \times 100 = 162.75$

Result: The Laspeyres Price Index for the current year is 162.75.

Interpretation:

1. The index number for the base year is taken as 100. Since the computed index is 162.75, the general price level in the current year is 62.75% higher than in the base year.
2. A basket of goods that could be bought for Rs 100 in the base year costs Rs 162.75 in the current year.
3. The purchasing power (value) of money has fallen to $\frac{100}{162.75} = 0.6144$, i.e. one rupee now buys goods worth only about 61 paise of the base year.
4. Since prices have risen faster than money income (if income is unchanged), the real income and standard of living of consumers have declined.
5. The rise is measured with base year quantities as weights, so the index shows the pure effect of price change on the base year consumption basket, without being disturbed by changes in the consumption pattern.