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CBSE class 11 commerce Economics Statistical Tools and Interpretation

CBSE class 11 commerce Economics Statistical Tools and Interpretation – CBSE Class 11 commerce economics notes for students according to NCERT syllabus will be provided by us. The syllabus includes Part A, Part B and Part C. Part A includes Statistics and Economics. Here we are providing important concepts about CBSE class 11 commerce Economics statistical Tools and Interpretation.

CBSE class 11 commerce Economics Statistical Tools and Interpretation

CBSE class 11 commerce Economics Statistical Tools and Interpretation unit 3 detailed concepts

Unit 3: Statistical Tools and Interpretation

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CBSE class 11 commerce Economics Statistical Tools and Interpretation – (For all the numerical problems and solutions, the appropriate economic interpretation may be attempted. This means, the students need to solve the problems and provide interpretation for the results derived.) Measures of Central Tendency

  • Measures of Central Tendency – mean (simple and weighted), median and mode
  • Measures of Dispersion – absolute dispersion (range, quartile deviation, mean deviation and standard deviation); relative dispersion (co-efficient of quartile-deviation, co-efficient of mean deviation, co-efficient of variation); Lorenz Curve: Meaning and its application.
  • Correlation – meaning, scatter diagram; Measures of correlation – Karl Pearson’s method (two variables ungrouped data) Spearman’s rank correlation.
  • Introduction to Index Numbers – meaning, types – wholesale price index, consumer price index and index of industrial production, uses of index numbers; Inflation and index numbers.

Use of Statistical Tools and Interpretation

CBSE class 11 commerce Economics Statistical Tools and Interpretation

CBSE class 11 commerce Economics Statistical Tools and Interpretation

Measures of Central Tendency

Introduction

A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean (often called the average) is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.

The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used.

Central Tendency: The Mean Vector

Throughout this course, we’ll use the ordinary notations for the mean of a variable. That is, the symbol μ is used to represent a (theoretical) population mean and the symbol x¯ is used to represent a sample mean computed from observed data. In the multivariate setting, we add subscripts to these symbols to indicate the specific variable for which the mean is being given. For instance, μ1 represents the population mean for variable x1 and x¯1 denotes a sample mean based on observed data for variable x¯1.

The population mean is the measure of central tendency for the population. Here, the population mean for variable j is

μj=E(Xij)

The notation E stands for statistical expectation; here E(Xij) is the mean of Xij over all members of the population, or equivalently, over all random draws from a stochastic model. For example, μj=E(Xij) may be the mean of a normal variable.

The population mean μj for variable j can be estimated by the sample mean

x¯j=1n∑i=1nXij

Note: the sample mean x¯j, because it is a function of our random data is also going to have a mean itself. In fact, the population mean of the sample mean is equal to population mean μj; i.e.,

E(x¯j)=μj

Therefore, the x¯j is unbiased for μj.

Another way of saying this is that the mean of the x¯j’s over all possible samples of size n is equal to μj.

Recall that the population mean vector is μ which is a collection of the means for each of the population means for each of the different variables.

μ=⎛⎝⎜⎜⎜⎜μ1μ2⋮μp⎞⎠⎟⎟⎟⎟

We can estimate this population mean vector, μ, by x¯. This is obtained by collecting the sample means from each of the variables in a single vector. This is shown below.

x¯=⎛⎝⎜⎜⎜⎜x¯1x¯2⋮x¯p⎞⎠⎟⎟⎟⎟=⎛⎝⎜⎜⎜⎜⎜1n∑ni=1Xi11n∑ni=1Xi2⋮1n∑ni=1Xip⎞⎠⎟⎟⎟⎟⎟=1n∑i=1nXi

Just as the sample means, x¯, for the individual variables are unbiased for their respective population means, note that the sample mean vectors is unbiased for the population mean vectors.

E(x¯)=E⎛⎝⎜⎜⎜⎜x¯1x¯2⋮x¯p⎞⎠⎟⎟⎟⎟=⎛⎝⎜⎜⎜⎜E(x¯1)E(x¯2)⋮E(x¯p)⎞⎠⎟⎟⎟⎟=⎛⎝⎜⎜⎜⎜μ1μ2⋮μp⎞⎠⎟⎟⎟⎟=μ

Measures of Dispersion

CBSE class 11 commerce Economics Statistical Tools and Interpretation – Dispersion in statistics is a way of describing how spread out a set of data is. When a data set has a large value, the values in the set are widely scattered; when it is small the items in the set are tightly clustered. Very basically, this set of data has a small value:
1, 2, 2, 3, 3, 4
…and this set has a wider one:
0, 1, 20, 30, 40, 100

Dispersion: Variance, Standard Deviation

CBSE class 11 commerce Economics Statistical Tools and Interpretation – A variance measures the degree of spread (dispersion) in a variable’s values. Theoretically, a population variance is the average squared difference between a variable’s values and the mean for that variable. The population variance for variable xj is

σ2j=E(xj−uj)2

Note that the squared residual (Xij−μj)2 is a function of the random variable Xij. Therefore, the squared residual itself is random, and has a population mean. The population variance is thus the population mean of the squared residual. We see that if the data tend to be far away from the mean, the squared residual will tend to be large, and hence the population variance will also be large. Conversely, if the the data tend to be close to the mean, the squared residual will tend to be small, and hence the population variance will also be small.

The population variance σ2j can be estimated by the sample variance

s2j=1n−1∑i=1n(Xij−x¯j)2=∑ni=1X2ij−((∑ni=1Xij)2/n)n−1

The first expression in this formula is most suitable for interpreting the sample variance. We see that it is a function of the squared residuals; that is, take difference between the individual observations and their sample mean, and then square the result. Here, we may observe that if tend to be far away from their sample means, then the squared residuals and hence the sample variance will also tend to be large.

If on the other hand, if the observations tend to be close to their respective sample means, then the squared differences between the data and their means will be small, resulting is a small sample variance value for that variable.

The last part of the expression above, gives the formula that is most suitable for computation, either by hand or by a computer! Since the sample variance is a function of the random data, the sample variance itself is a random quantity, and so has a population mean. In fact, the population mean of the sample variance is equal to the population variance:

E(s2j)=σ2j

That is, the sample variance s2j is unbiased for the population variance σ2j.

Our textbook (Johnson and Wichern, 6th ed.) uses a sample variance formula derived using maximum likelihood estimation principles. In this formula, the division is by n rather than n−1.

s2j=∑ni=1(xij−x¯j)2n

Correlation

CBSE class 11 commerce Economics Statistical Tools and Interpretation – Correlation is a statistical technique that can show whether and how strongly pairs of variables are related. For example, height and weight are related; taller people tend to be heavier than shorter people. The relationship isn’t perfect. People of the same height vary in weight, and you can easily think of two people you know where the shorter one is heavier than the taller one. Nonetheless, the average weight of people 5’5” is less than the average weight of people 5’6”, and their average weight is less than that of people 5’7”, etc. Correlation can tell you just how much of the variation in peoples’ weights is related to their heights.

Techniques in Determining Correlation

There are several different correlation techniques. The Survey System’s optional Statistics Module includes the most common type, called the Pearson or product-moment correlation. The module also includes a variation on this type called partial correlation. The latter is useful when you want to look at the relationship between two variables while removing the effect of one or two other variables.

Like all statistical techniques, correlation is only appropriate for certain kinds of data. Correlation works for quantifiable data in which numbers are meaningful, usually quantities of some sort. It cannot be used for purely categorical data, such as gender, brands purchased, or favorite color.

Introduction to Index Numbers

Index numbers are commonly used statistical device for measuring the combined fluctuations in a group related variables. If we wish to compare the price level of consumer items today with that prevalent ten years ago, we are not interested in comparing the prices of only one item, but in comparing some sort of average price levels. We may wish to compare the present agricultural production or industrial production with that at the time of independence. Here again, we have to consider all items of production and each item may have undergone a different fractional increase (or even a decrease). How do we obtain a composite measure? This composite measure is provided by index numbers which may be defined as a device for combining the variations that have come in group of related variables over a period of time, with a view to obtain a figure that represents the ‘net’ result of the change in the constitute variables.

Index numbers are statistical measures designed to show changes in a variable or group of related variables with respect to time, geographic location or other characteristics such as income, profession, etc. A collection of index numbers for different years, locations, etc., is sometimes called an index series.

Simple Index Number:

Composite Index Number:

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CBSE class 11 commerce Economics Part A Statistics for Economics Ch1 notes

CBSE class 11 commerce Economics Part A Statistics for Economics Ch2 notes

CBSE class 11 commerce Economics Part A Statistics for Economics Ch3 notes

CBSE class 11 commerce Economics Part A Statistics for Economics Ch4 notes

CBSE class 11 commerce Economics Part A Statistics for Economics Ch5 notes

CBSE class 11 commerce Economics Part A Statistics for Economics Ch6 notes

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CBSE class 11 commerce Economics Part A Statistics for Economics Ch9 notes

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