# Chapter 3 Descriptive Statistics And Data Visualization

Content

Outliers can be problematic when you want to summarize a sample using an average. They will come up again when we talk about correlation.

When this is the case, outlier detection falls prey to predictable inaccuracies—it detects outliers far more often. We can eliminate the units of measurement by dividing the residuals by an estimate of their standard deviation, thereby obtaining what are known as standardized residuals.

Below are some tips that can help to make handling them a little bit less elusive. It looks like the potential flagged outliers are pretty uniformly distributed. In addition to being a marketing research consultant, he has been published in several academic journals and trade publications and taught post-graduate students. He also serves as an editorial reviewer for marketing journals. In his spare time, he travels and publishes GlobeRovers Magazine for intrepid travellers, and has also published 10 books. Bell-shaped distributions are studied further later (for example, Chap. 17). The larger the variance, the more spread apart the numbers are, on average.

In addition, most major testing tools have strategies for dealing with outliers, but they usually differ in how they do so. Ø Below three statistical tests use the concept of hypothesis testing to identify outliers.

## Dont Drop An Outlier If:

To explain the types of missing values, the following elements are defined under the assumption that the total number of study participants is ‘i’ and the total number of measurements is ‘j’. Missing values and outliers are frequently encountered during the data collection phase of observational or experimental studies conducted in all fields of natural and social sciences. Finally, let’s revisit the model we used at the start of this section, identifying outliers in spss predicting api00from meals, ell and emer. Using this model, the distribution of the residuals looked very nice and even across the fitted values. Will this automatically ruin the distribution of the residuals? Now, let’s run the analysis omitting DC by using the filter command to omit “dc” from the analysis. As we expect, deleting DC made a large change in the coefficient for single .The coefficient for singledropped from 132.4 to 89.4.

The table in the output will show the number of missing values for each variable. It has detailed univariate descriptive statistics for each of the continuous variables, including skewness and kurtosis.

## How Are Outliers Determined In Statistics?

In these cases we can take the steps from above, changing only the number that we multiply the IQR by, and define a certain type of outlier. If we subtract 3.0 x IQR from the first quartile, any point that is below this number is called a strong outlier. In the same way, the addition of 3.0 x IQR to the third quartile allows us to define strong outliers by looking at points which are greater than this number. Examining data for outliers is a common first step in analysing data.

- An outlier is a value that is very different from the other data in your data set.
- The specified number of standard deviations is called the threshold.
- The aim, therefore, is to test to see if there are any outliers in this dataset and to remove them.
- The weight modification method allows weight modification without discarding or replacing the values of outliers, limiting the influence of the outliers.
- As for most of data analysis, using common sense is usually a better idea…
- Ignoring them can skew your data or make you miss a problem you might not have otherwise expected.

If the outlier turns out to be a result of a data entry error, you may decide to assign a new value to it such as the mean or the median of the dataset. Set your range for what’s valid , and consistently delete any data points outside of the range. Using this information, sketch the boxplot and the histogram for the data.

## How Do You Know When To Remove Outliers?

So, say you have a mean that differs quite a bit from the median, it probably means you have some very large or small values skewing it. “On average, what a customer spends is not normally distributed. Even though this has a little cost, filtering out outliers is worth it. You often discover significant effects that are simply “hidden” by outliers. Data visualization is a core discipline for analysts and optimizers, not just to better communicate results with executives, but to explore the data fully. The answer could differ from business to business, but it’s important to have the conversation rather than ignore the data, regardless of the significance. Ø Whereas Chi-square test can be used for the same with the chi-square distribution.

Although a looser rule is an overall kurtosis score of 2.200 or less (rather than 1.00) (Sposito et al., 1983). In general, you should never rely on the results of these hypothesis tests alone; you should always use other sources to support or refute a claim of normality. You should also consider your sample size and any scientific background about your problem that is relevant. Keep in mind that the Shapiro-Wilk test is very sensitive to even trivial deviations in normality when the sample size is large. That is, if your sample size is very large, it is quite possible that the Shapiro-Wilk test may come back significant, even if the deviations from normality are very small. In SPSS, the Explore procedure produces univariate descriptive statistics, as well as confidence intervals for the mean, normality tests, and plots. In each case, the difference is calculated between historical data points and values calculated by the various forecasting methods.

Then you can make a conclusion based on what is best for your set of data. For example, the standard deviation rule is sometimes given slightly differently; for example, outliers identified as observations more than 2.5 standard deviations away from the mean.

Percentile is the percent of scores in a distribution at or below a particular score. You can plot the frequencies on the y-axis and the scores along the x-axis to create a graph called a histogram. For our state anxiety measure, “78” would be listed on the x-axis and have a bar height of 2. “79” would be listed next to it and have a bar height of 1. At its core, it belongs to the resampling methods, which provide reliable estimates of the distribution of variables on the basis of the observed data through random sampling procedures. In another section of Dr. Julia Engelmann’s wonderful article for our blog, she shared a graphic depicting this difference. The left graphic shows a perfect normal distribution.

An outlier may be due to variability in the measurement or it may indicate experimental error; the latter are sometimes excluded from the data set. An outlier can cause serious problems in statistical analyses.

Since DC is really not a state, we can use this to justify omitting it from the analysis saying that we really wish to just analyze states. Note that the regression line is not automatically produced in the graph. We double clicked on the graph, and then chose “Chart” and the “Options” and then chose “Fit Line Total” to add a regression line to each of the graphs below.

This measure is called DFBETA and a DFBETA value can be computed for each observation for each predictor. As shown below, we use the /save sdbeta subcommand to save the DFBETA values for each of the predictors. This saves 4 variables into the current data file, sdfb1, sdfb2, sdfb3 and sdfb4, corresponding to the DFBETA for the Intercept and for pctmetro, povertyand for single, respectively.

Outliers can be very informative about the subject-area and data collection process. It’s essential to understand how outliers occur and whether they might happen again as a normal part of the process or study area.

## When To Drop Or Keep Outliers

The different approaches for handling missing values and outliers can drastically change the results of data analysis. Therefore, adequate treatment of missing data and outliers is crucial for analysis. In this review paper, we discuss the types of missing values and different methods used to identify outliers and to handle missing values and outliers efficiently.

- Using these rules, we can see from the table below, that all three variables are fine using the first rule, but using the second rule, they are all negative skewed.
- During his tenure, he has worked with global clients in various domains like Banking, Insurance, Private Equity, Telecom and Human Resource.
- These plots are useful for seeing how a single point may be influencing the regression line, while taking other variables in the model into account.
- For all the attention we give normality, deviations from normality often do not have much impact on your statistical decision.

As we have seen, DC is an observation that both has a large residual and large leverage. We can make a plot that shows the leverage by the residual and look for observations that are high in leverage and have a high residual. We can do this using the /scatterplot subcommand as shown below. This is a quick way of checking potential influential observations and outliers at the same time.

## Detecting Multivariate Outliers

When you have extremely non-normal data, it will influence your regressions in SPSS and AMOS. In such cases, if you have non-Likert-scale variables (so, variables like age, income, revenue, etc.), you can transform them prior to including them in your model.

- Remember, the curve is not the actual data; it is where the data would have been if the distribution was perfectly normal.
- Other times, though, they can influence a data set, so it’s important to keep them to better understand the big picture.
- Dealing with outliers is essential prior to the analysis of the data set containing outlier.
- In the Summary tab, the method used and the number of outliers will be recorded.
- The larger the sample size, the less the impact of skewness and kurtosis.

In optimization, most outliers are on the higher end because of bulk orderers. Given your knowledge of historical data, if you’d like to do a post-hoc trimming of values above a certain parameter, that’s easy to do in R.

## Outliers

When you have selected the outlier method and specified how aggressive you will be in their removal, click the ‘OK‘ button to run the test. For this example, I have a set of 22 replicates stacked into a column data sheet. Outliers are data points which are distinctly different from the rest. The definition can vary depending upon the method used.

This approach involves modifying the weights of outliers or replacing the values being tested for outliers with expected values. The weight modification method allows weight modification without discarding or replacing the https://business-accounting.net/ values of outliers, limiting the influence of the outliers. The value modification method allows the replacement of the values of outliers with the largest or second smallest value in observations excluding outliers.

Multivariate outliers can be a tricky statistical concept for many students. Multivariate outliers are typically examined when running statistical analyses with two or more independent or dependent variables. Here we outline the steps you can take to test for the presence of multivariate outliers in SPSS.

After having deleted DC, we would repeat the process we have illustrated in this section to search for any other outlying and influential observations. We have shown a few examples of the variables that you can refer to in the /residuals, /casewise, /scatterplot and /save sdbeta subcommands. Here is a list of all of the variables that can be used on these subcommands; however, not all variables can be used on each subcommand. The following table summarizes the general rules of thumb we use for the measures we have discussed for identifying observations worthy of further investigation .

Note that each frequency table only contains a handful of outliers for which |z| ≥ 3.29. We’ll now exclude these values from all data analyses and editing with the syntax below.

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