Ch1 - Chapter 1 Univariate Distributions 1 Descriptive...

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1 This sample is drawn from the Survey of Consumer Finance (SCF), 1996. This large survey questionnaire was completed by almost 100,000 adult Canadians and provides information on sources of income, hours of work and family characteristics during 1995. Further information about the SCF is available at The subsample of 3,921 individuals used here is a random subset of the full sample. Some restrictions were imposed when the sample was drawn. In particular, only workers who stated they worked full-time throughout the year were included. 2 Refer to the appendix of this chapter for details on the properties of the summation operator, Chapter 1 Univariate Distributions 1 Descriptive Statistics The most basic application of statistical concepts is to describe data. In many situations large quantities of data are available to researchers and typically, the most urgent problem is to find a way of presenting the data so that the most important features can be highlighted. One useful approach is to construct a diagram known as a histogram for each variable. Figure 1.1 is a histogram that was constructed from 3,921 observations on the hourly pay earned by full-time Canadian workers in 1995. 1 The data have been sorted into 10 bins. The centre of each bin is recorded on the horizontal axis. For example, the first bin contains all the wage rates in the sample that lie between $2.00 and $6.00 per hour - its centre is at $4.00 per hour. The number of observations within a bin is called the frequency and this type of histogram is known as a frequency distribution because it shows how the frequencies are distributed amongst the bins. Since each observation falls in only one bin, the sum of the frequencies is the sample size, 3,921. By rescaling the vertical axis, the heights of the bars in Figure 1.1 can also be interpreted as the relative frequencies , which are obtained by dividing each frequency by the sample size. For example, the relative frequency of the first bin is 177/3921 = 0.045 In other words, 4.5% of the sample falls in the first bin. Clearly, the sum of the relative frequencies (or shares) must be unity. It will be useful if some notation is used to refer to key concepts. The size of the entire sample is defined to be n (n = 3,921 in the example). The number of bins is m, where m < n and in the wage example m = 10. The frequency of observations in the j th bin is denoted by f j for j = 1, 2, . .., m. In the example, f 1 = 177. The sum of the frequencies must equal the total number of observations in the sample 2 :
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Econometrics Text by D M Prescott © Chapter 1, 2 f 1 ± f 2 ± .... ± f m ² M m j 1 f j ² n [1.1] f 1 n ± f 2 n ± .... ± f m n ² M m j 1 f j n ² 1 n M m j 1 f j ² n n ² 1 r 1 ± r 2 .... ± r m ² M m j 1 r j ² 1 0 0.05 0.1 0.15 0.2 0.25 Relative Frequency Wage Rates ($/hour) Distribution of Wages 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 68 Figure 1.1 0 0.01 0.02 0.03 0.04 0.05 0.06 Density Wage Rates ($/hour) Distribution of Wages 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 68 Figure 1.2 Density = Relative Freq / bin width
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This note was uploaded on 02/02/2012 for the course ECON 2410 taught by Professor Prescott during the Fall '11 term at University of Guelph.

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Ch1 - Chapter 1 Univariate Distributions 1 Descriptive...

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