Descriptive statistics can be manipulated in many ways that can be misleading. Effects of Changing Scale : In this graph, the earnings scale is greater. Effects of Changing Scale : This is a graph plotting yearly earnings. Both graphs plot the years , , and along the x-axis. Bias is another common distortion in the field of descriptive statistics. A statistic is biased if it is calculated in such a way that is systematically different from the population parameter of interest.
The following are examples of statistical bias. Descriptive statistics is a powerful form of research because it collects and summarizes vast amounts of data and information in a manageable and organized manner.
Moreover, it establishes the standard deviation and can lay the groundwork for more complex statistical analysis. In other words, every time you try to describe a large set of observations with a single descriptive statistics indicator, you run the risk of distorting the original data or losing important detail. Exploratory data analysis is an approach to analyzing data sets in order to summarize their main characteristics, often with visual methods.
Exploratory data analysis EDA is an approach to analyzing data sets in order to summarize their main characteristics, often with visual methods.
It is a statistical practice concerned with among other things :. Primarily, EDA is for seeing what the data can tell us beyond the formal modeling or hypothesis testing task. EDA is different from initial data analysis IDA , which focuses more narrowly on checking assumptions required for model fitting and hypothesis testing, handling missing values, and making transformations of variables as needed. Exploratory data analysis was promoted by John Tukey to encourage statisticians to explore the data and possibly formulate hypotheses that could lead to new data collection and experiments.
Both of these try to reduce the sensitivity of statistical inferences to errors in formulating statistical models. Tukey promoted the use of the five number summary of numerical data:. His reasoning was that the median and quartiles, being functions of the empirical distribution, are defined for all distributions, unlike the mean and standard deviation.
Moreover, the quartiles and median are more robust to skewed or heavy-tailed distributions than traditional summaries the mean and standard deviation. Such problems included the fabrication of semiconductors and the understanding of communications networks. These statistical developments, all championed by Tukey, were designed to complement the analytic theory of testing statistical hypotheses.
Tukey held that too much emphasis in statistics was placed on statistical hypothesis testing confirmatory data analysis and more emphasis needed to be placed on using data to suggest hypotheses to test. In particular, he held that confusing the two types of analyses and employing them on the same set of data can lead to systematic bias owing to the issues inherent in testing hypotheses suggested by the data.
Although EDA is characterized more by the attitude taken than by particular techniques, there are a number of tools that are useful.
Many EDA techniques have been adopted into data mining and are being taught to young students as a way to introduce them to statistical thinking. Typical graphical techniques used in EDA are:. These EDA techniques aim to position these plots so as to maximize our natural pattern-recognition abilities.
A clear picture is worth a thousand words! Privacy Policy. Skip to main content. Measures of Variation. Search for:. Describing Variability. Range The range is a measure of the total spread of values in a quantitative dataset. Learning Objectives Interpret the range as the overall dispersion of values in a dataset.
Key Takeaways Key Points Unlike other more popular measures of dispersion, the range actually measures total dispersion between the smallest and largest values rather than relative dispersion around a measure of central tendency. Because the information the range provides is rather limited, it is seldom used in statistical analyses.
The mid-range of a set of statistical data values is the arithmetic mean of the maximum and minimum values in a data set. Key Terms range : the length of the smallest interval which contains all the data in a sample; the difference between the largest and smallest observations in the sample dispersion : the degree of scatter of data.
Variance Variance is the sum of the probabilities that various outcomes will occur multiplied by the squared deviations from the average of the random variable.
Learning Objectives Calculate variance to describe a population. Key Terms deviation : For interval variables and ratio variables, a measure of difference between the observed value and the mean. Standard Deviation: Definition and Calculation Standard deviation is a measure of the average distance between the values of the data in the set and the mean.
Learning Objectives Contrast the usefulness of variance and standard deviation. Key Takeaways Key Points A low standard deviation indicates that the data points tend to be very close to the mean; a high standard deviation indicates that the data points are spread out over a large range of values. In addition to expressing the variability of a population, standard deviation is commonly used to measure confidence in statistical conclusions.
To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result of each. Next, compute the average of these values, and take the square root. Key Terms normal distribution : A family of continuous probability distributions such that the probability density function is the normal or Gaussian function. Interpreting the Standard Deviation The practical value of understanding the standard deviation of a set of values is in appreciating how much variation there is from the mean.
Learning Objectives Derive standard deviation to measure the uncertainty in daily life examples. Key Takeaways Key Points A large standard deviation indicates that the data points are far from the mean, and a small standard deviation indicates that they are clustered closely around the mean.
In finance, standard deviation is often used as a measure of the risk associated with price-fluctuations of a given asset stocks, bonds, property, etc. Key Terms standard deviation : a measure of how spread out data values are around the mean, defined as the square root of the variance disparity : the state of being unequal; difference. Using a Statistical Calculator For advanced calculating and graphing, it is often very helpful for students and statisticians to have access to statistical calculators.
Key Terms TI : A calculator manufactured by Texas Instruments that is one of the most popular graphing calculators for statistical purposes. R : A free software programming language and a software environment for statistical computing and graphics. Degrees of Freedom The number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Key Takeaways Key Points The degree of freedom can be defined as the minimum number of independent coordinates which can specify the position of the system completely.
A parameter is a characteristic of the variable under examination as a whole; it is part of describing the overall distribution of values. As more degrees of freedom are lost, fewer and fewer different situations are accounted for by a model since fewer and fewer pieces of information could, in principle, be different from what is actually observed. Key Terms residual : The difference between the observed value and the estimated function value.
Interquartile Range The interquartile range IQR is a measure of statistical dispersion, or variability, based on dividing a data set into quartiles. Learning Objectives Calculate interquartile range based on a given data set. Key Terms outlier : a value in a statistical sample which does not fit a pattern that describes most other data points; specifically, a value that lies 1. Measures of Variability of Qualitative and Ranked Data Variability for qualitative data is measured in terms of how often observations differ from one another.
Learning Objectives Assess the use of IQV in measuring statistical dispersion in nominal distributions. Instead, we should focus on the unlikeability, or how often observations differ. An index of qualitative variation IQV is a measure of statistical dispersion in nominal distributions —or those dealing with qualitative data.
The variation ratio is the simplest measure of qualitative variation. It is defined as the proportion of cases which are not the mode. Key Terms variation ratio : the proportion of cases not in the mode qualitative data : data centered around descriptions or distinctions based on some quality or characteristic rather than on some quantity or measured value.
Distorting the Truth with Descriptive Statistics Descriptive statistics can be manipulated in many ways that can be misleading, including the changing of scale and statistical bias. Learning Objectives Assess the significance of descriptive statistics given its limitations.
Key Takeaways Key Points Descriptive statistics is a powerful form of research because it collects and summarizes vast amounts of data and information in a manageable and organized manner.
Descriptive statistics, however, lacks the ability to identify the cause behind the phenomenon, correlate associate data, account for randomness, or provide statistical calculations that can lead to hypothesis or theories of populations studied. Every time you try to describe a large set of observations with a single descriptive statistics indicator, you run the risk of distorting the original data or losing important detail. Key Terms bias : Uncountable Inclination towards something; predisposition, partiality, prejudice, preference, predilection.
Exploratory Data Analysis EDA Exploratory data analysis is an approach to analyzing data sets in order to summarize their main characteristics, often with visual methods. Key Takeaways Key Points EDA is concerned with uncovering underlying structure, extracting important variables, detecting outliers and anomalies, testing underlying assumptions, and developing models. Robust statistics and nonparametric statistics both try to reduce the sensitivity of statistical inferences to errors in formulating statistical models.
Key Terms exploratory data analysis : an approach to analyzing data sets that is concerned with uncovering underlying structure, extracting important variables, detecting outliers and anomalies, testing underlying assumptions, and developing models data mining : a technique for searching large-scale databases for patterns; used mainly to find previously unknown correlations between variables that may be commercially useful skewed : Biased or distorted pertaining to statistics or information.
Hope that proves useful. Conducting successful research requires choosing the appropriate study design. This article describes the most common types of designs conducted by researchers.
What are the key steps in EBM? Who are S4BE? Eveliina Ilola View more posts from Eveliina. Leave a Reply Cancel reply Your email address will not be published. Terje Soerdal Very simply and nicely explained. Sayyid excellent explanation of the concepts 5th November at pm Reply to Sayyid.
Mustapha How do you then determine the sample size with the most minimal acceptable standard error. Osama Elbahr Thanks for your illustrations. But, can you clarify when to incorporate SE in our research results and how to interpret?
Thanks again 19th February at pm Reply to Osama. Guru I was not able to understand standard error. Rohit Standard Deviation is the square root of variance, so its kind of trivial to state the conclusion about the increasing standard error with respect to standard error. It is a good write up 6th October at pm Reply to Rohit. Emma Carter Thank you for flagging the symbol errors on the page Rohit.
A t-test is a statistical test that compares the means of two samples. It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero. Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test.
Significance is usually denoted by a p -value , or probability value. Statistical significance is arbitrary — it depends on the threshold, or alpha value, chosen by the researcher. When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.
A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups. The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis.
Different test statistics are used in different statistical tests. The measures of central tendency you can use depends on the level of measurement of your data. Ordinal data has two characteristics:. Nominal and ordinal are two of the four levels of measurement. Nominal level data can only be classified, while ordinal level data can be classified and ordered.
If your confidence interval for a difference between groups includes zero, that means that if you run your experiment again you have a good chance of finding no difference between groups. If your confidence interval for a correlation or regression includes zero, that means that if you run your experiment again there is a good chance of finding no correlation in your data. In both of these cases, you will also find a high p -value when you run your statistical test, meaning that your results could have occurred under the null hypothesis of no relationship between variables or no difference between groups.
If you want to calculate a confidence interval around the mean of data that is not normally distributed , you have two choices:. The standard normal distribution , also called the z -distribution, is a special normal distribution where the mean is 0 and the standard deviation is 1.
Any normal distribution can be converted into the standard normal distribution by turning the individual values into z -scores. In a z -distribution, z -scores tell you how many standard deviations away from the mean each value lies. The z -score and t -score aka z -value and t -value show how many standard deviations away from the mean of the distribution you are, assuming your data follow a z -distribution or a t -distribution.
These scores are used in statistical tests to show how far from the mean of the predicted distribution your statistical estimate is. If your test produces a z -score of 2. The predicted mean and distribution of your estimate are generated by the null hypothesis of the statistical test you are using. The more standard deviations away from the predicted mean your estimate is, the less likely it is that the estimate could have occurred under the null hypothesis.
To calculate the confidence interval , you need to know:. Then you can plug these components into the confidence interval formula that corresponds to your data. The formula depends on the type of estimate e. The confidence level is the percentage of times you expect to get close to the same estimate if you run your experiment again or resample the population in the same way. The confidence interval is the actual upper and lower bounds of the estimate you expect to find at a given level of confidence.
These are the upper and lower bounds of the confidence interval. Nominal data is data that can be labelled or classified into mutually exclusive categories within a variable.
These categories cannot be ordered in a meaningful way. For example, for the nominal variable of preferred mode of transportation, you may have the categories of car, bus, train, tram or bicycle.
The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average. Statistical tests commonly assume that:. If your data does not meet these assumptions you might still be able to use a nonparametric statistical test , which have fewer requirements but also make weaker inferences. Measures of central tendency help you find the middle, or the average, of a data set. Some variables have fixed levels.
For example, gender and ethnicity are always nominal level data because they cannot be ranked. However, for other variables, you can choose the level of measurement. For example, income is a variable that can be recorded on an ordinal or a ratio scale:. If you have a choice, the ratio level is always preferable because you can analyze data in more ways. The higher the level of measurement, the more precise your data is.
The level at which you measure a variable determines how you can analyze your data. Depending on the level of measurement , you can perform different descriptive statistics to get an overall summary of your data and inferential statistics to see if your results support or refute your hypothesis.
Levels of measurement tell you how precisely variables are recorded. There are 4 levels of measurement, which can be ranked from low to high:. The p -value only tells you how likely the data you have observed is to have occurred under the null hypothesis. The alpha value, or the threshold for statistical significance , is arbitrary — which value you use depends on your field of study.
In most cases, researchers use an alpha of 0. P -values are usually automatically calculated by the program you use to perform your statistical test. They can also be estimated using p -value tables for the relevant test statistic. P -values are calculated from the null distribution of the test statistic. They tell you how often a test statistic is expected to occur under the null hypothesis of the statistical test, based on where it falls in the null distribution. If the test statistic is far from the mean of the null distribution, then the p -value will be small, showing that the test statistic is not likely to have occurred under the null hypothesis.
A p -value , or probability value, is a number describing how likely it is that your data would have occurred under the null hypothesis of your statistical test. You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test. The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.
For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis, even if the true correlation between two variables is the same in either data set.
Want to contact us directly? No problem. We are always here for you. Scribbr specializes in editing study-related documents. We proofread:. You can find all the citation styles and locales used in the Scribbr Citation Generator in our publicly accessible repository on Github. Frequently asked questions See all. Home Frequently asked questions What does standard deviation tell you? What does standard deviation tell you? Frequently asked questions: Statistics How do I find the median?
Can there be more than one mode? Your data can be: without any mode unimodal, with one mode, bimodal, with two modes, trimodal, with three modes, or multimodal, with four or more modes. How do I find the mode? To find the mode : If your data is numerical or quantitative, order the values from low to high. If it is categorical, sort the values by group, in any order. Then you simply need to identify the most frequently occurring value. When should I use the interquartile range?
What are the two main methods for calculating interquartile range? What is homoscedasticity? What is variance used for in statistics? Both measures reflect variability in a distribution, but their units differ: Standard deviation is expressed in the same units as the original values e.
Variance is expressed in much larger units e. What is the empirical rule? Around What is a normal distribution? When should I use the median? Can the range be a negative number? What is the range in statistics?
What are the 4 main measures of variability? Variability is most commonly measured with the following descriptive statistics : Range : the difference between the highest and lowest values Interquartile range : the range of the middle half of a distribution Standard deviation : average distance from the mean Variance : average of squared distances from the mean. What is variability? Variability is also referred to as spread, scatter or dispersion.
What is the difference between interval and ratio data? What is a critical value? What is the difference between the t-distribution and the standard normal distribution? What is a t-score? What is a t-distribution? Is the correlation coefficient the same as the slope of the line? What do the sign and value of the correlation coefficient tell you? What are the assumptions of the Pearson correlation coefficient?
What is a correlation coefficient? How do you increase statistical power? There are various ways to improve power: Increase the potential effect size by manipulating your independent variable more strongly, Increase sample size, Increase the significance level alpha , Reduce measurement error by increasing the precision and accuracy of your measurement devices and procedures, Use a one-tailed test instead of a two-tailed test for t tests and z tests.
What is a power analysis? Sample size : the minimum number of observations needed to observe an effect of a certain size with a given power level. Expected effect size : a standardized way of expressing the magnitude of the expected result of your study, usually based on similar studies or a pilot study. What are null and alternative hypotheses? What is statistical analysis? How do you reduce the risk of making a Type II error?
How do you reduce the risk of making a Type I error? To reduce the Type I error probability, you can set a lower significance level.
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