Exploratory Data Analysis
1.4. EDA Case Studies
1.4.2. Case Studies
220.127.116.11. Normal Random Numbers
As a first step in the analysis, common summary statistics are
computed from the data.
Sample size = 500 Mean = -0.2935997E-02 Median = -0.9300000E-01 Minimum = -0.2647000E+01 Maximum = 0.3436000E+01 Range = 0.6083000E+01 Stan. Dev. = 0.1021041E+01
One way to quantify a change in location over time is to
fit a straight line
to the data using an index variable as the independent
variable in the regression. For our data, we assume
that data are in sequential run order and that the
data were collected at equally spaced time intervals. In our regression,
we use the index variable X = 1, 2, ..., N, where N is the number
of observations. If there is no significant drift in the location
over time, the slope parameter should be zero.
Coefficient Estimate Stan. Error t-Value B0 0.699127E-02 0.9155E-01 0.0764 B1 -0.396298E-04 0.3167E-03 -0.1251 Residual Standard Deviation = 1.02205 Residual Degrees of Freedom = 498The absolute value of the t-value for the slope parameter is smaller than the critical value of t0.975,498 = 1.96. Thus, we conclude that the slope is not different from zero at the 0.05 significance level.
One simple way to detect a change in variation is with
Bartlett's test, after dividing
the data set into several equal-sized intervals.
The choice of the number of intervals is somewhat arbitrary, although
values of four or eight are reasonable. We will divide our data into
H0: σ12 = σ22 = σ32 = σ42 Ha: At least one σi2 is not equal to the others. Test statistic: T = 2.373660 Degrees of freedom: k - 1 = 3 Significance level: α = 0.05 Critical value: Χ21-α,k-1 = 7.814728 Critical region: Reject H0 if T > 7.814728In this case, Bartlett's test indicates that the variances are not significantly different in the four intervals.
There are many ways in which data can be non-random. However,
most common forms of non-randomness can be detected with a
few simple tests including the
lag plot shown on
the previous page.
Another check is an autocorrelation plot that shows the autocorrelations for various lags. Confidence bands can be plotted at the 95 % and 99 % confidence levels. Points outside this band indicate statistically significant values (lag 0 is always 1).
The lag 1 autocorrelation, which is generally the one of most interest, is 0.045. The critical values at the 5% significance level are -0.087 and 0.087. Since 0.045 is within the critical region, the lag 1 autocorrelation is not statistically significant, so there is no evidence of non-randomness.
A common test for randomness is the runs test.
H0: the sequence was produced in a random manner Ha: the sequence was not produced in a random manner Test statistic: Z = -1.0744 Significance level: α = 0.05 Critical value: Z1-α/2 = 1.96 Critical region: Reject H0 if |Z| > 1.96The runs test fails to reject the null hypothesis that the data were produced in a random manner.
are a graphical test for assessing if a
particular distribution provides an adequate fit to a data
A quantitative enhancement to the probability plot is the correlation coefficient of the points on the probability plot, or PPCC. For this data set the PPCC based on a normal distribution is 0.996. Since the PPCC is greater than the critical value of 0.987 (this is a tabulated value), the normality assumption is not rejected.
Chi-square and Kolmogorov-Smirnov goodness-of-fit tests are alternative methods for assessing distributional adequacy. The Wilk-Shapiro and Anderson-Darling tests can be used to test for normality. The results of the Anderson-Darling test follow.
H0: the data are normally distributed Ha: the data are not normally distributed Adjusted test statistic: A2 = 1.0612 Significance level: α = 0.05 Critical value: 0.787 Critical region: Reject H0 if A2 > 0.787The Anderson-Darling test rejects the normality assumption at the 0.05 significance level.
A test for outliers is the Grubbs
H0: there are no outliers in the data Ha: the maximum value is an outlier Test statistic: G = 3.368068 Significance level: α = 0.05 Critical value for an upper one-tailed test: 3.863087 Critical region: Reject H0 if G > 3.863087For this data set, Grubbs' test does not detect any outliers at the 0.05 significance level.
Since the underlying assumptions were validated both graphically
and analytically, we conclude that a reasonable model for the
It is sometimes useful and convenient to summarize the above results
in a report.
Analysis of 500 normal random numbers 1: Sample Size = 500 2: Location Mean = -0.00294 Standard Deviation of Mean = 0.045663 95% Confidence Interval for Mean = (-0.09266,0.086779) Drift with respect to location? = NO 3: Variation Standard Deviation = 1.021042 95% Confidence Interval for SD = (0.961437,1.088585) Drift with respect to variation? (based on Bartletts test on quarters of the data) = NO 4: Data are Normal? (as tested by Normal PPCC) = YES (as tested by Anderson-Darling) = NO 5: Randomness Autocorrelation = 0.045059 Data are Random? (as measured by autocorrelation) = YES 6: Statistical Control (i.e., no drift in location or scale, data are random, distribution is fixed, here we are testing only for fixed normal) Data Set is in Statistical Control? = YES 7: Outliers? (as determined by Grubbs' test) = NO