Chemical Engineering 426
Spring, 1997
B. Bernstein, Instructor
Assignment 2

 
Read:Sections 3-1, 3-2, 3-3

Problems:page 95 # 1, 2, 3, 4, 5, 7

Spring, 1997Chemical Engineering 426, Assignment 2page The following terms figure importantly in our subject:

Graphical Treatments of data provide ways of organizing them so that one can see trends and make interpretations. The following are useful graphical treatments of quantitative data:

Dot diagrams

Let x be the parameter measured. The values of x are rounded off to an agreed amount. Above each of the rounded off values on the x-axis are plotted the number of dots equal to the number of times that the measurements produced that value. (page 59)

Stem and leaf plot

Suppose that data, given to a certain number of places, say k places. One chooses positive integers m and k such that m+k=n and lists vertically the numbers m in order without omissions. Then, at each m for which there are data values, one lists the remaining k places horizontally for each data value measured. (page 60)

Back to back Stem and Leaf Plot

This is used to compare two sets of data with the m places listed vertically and the remaining k places listed to the right for one set of data and to the left for the other set of data. (page 61)

Histogram

Data are grouped into equal intervals and a at each such interval a bar is drawn whose width is that of the interval and whose height is the number of measured data points falling in that interval. (page 62) One may observe data possiblly bell shaped, right skewed, left skewed, uniform, bimodal, truncated, etc. (page 63)

Scatter plot

Bivariate data are plotted as points in a plane. (page 65)

The notion of quantile, quantile plots and Q-Q, or quantile-quantile plots figure importantly in understanding data. To this end, consider univariate quantative data arranged in increading order, . If p is a fraction, 0<p<1, the idea of the quantile is roughly as follows: Let x be such that p is the fraction of data points with valuew : Then x is the quantile, x=Q(p). The specific scheme which we shall use is that if p=(i-0.5)/n, then is the quantile and the other quantiles are found by linear interpolation. (page 67)

Median of a Distribution

Q(0.5)

First quartile

Q(0.25)

Third quartile

Q(0.75)

Deciles

Percentiles

Quantile plot

A plot of Q(p) versus p

Interquartile range

IQR=Q(0.75)-Q(0.25)

Box plot

A box is drawn from Q(0.25) to Q(0.75) with lines (whiskers) sticking out to reach the highest and the lowest data value within of the mean. Points extending further are plotted individually.

Q-Q plots

A quantile-quantile plot is a two dimensional graph for comparing two sets of data (or data against a theoretical distribution): Eanch point corresponds th the same quantile, but has an x-coordinate from one set of data and a y- coordinate from the other set of data. If the points fall on a straight line, the distributions are taken to be similar, or linearly related.

There are some calculated numerical parameters which figure in the understanding of data:

Sample Mean

For a sample, the sample mean is

Range

For the sample , the range,

Sample variance

(See equation 3-3) The sample variance, is

The quantity, is called the sample standard deviation.

Chebyshev's Theorem

If k>1, at least of the data are within ks of (page 83)

Population mean

If is the entire population, the population mean is

Population variance

The quantity is called the population standard deviation.

Taking samples at various times during a process and making plots of and R over time can give information on how the process is running and whether the machinery needs adjustment.

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