A measure of central tendency is a single value that attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location. They are also classed as summary statistics. The mean often called the average is most likely the measure of central tendency that you are most familiar with, but there are others, such as the median and the mode.
The mean, median and mode are all valid measures of central tendency, but under different conditions, some measures of central tendency become more appropriate to use than others. In the following sections, we will look at the mean, mode and median, and learn how to calculate them and under what conditions they are most appropriate to be used. The mean or average is the most popular and well known measure of central tendency.
It can be used with both discrete and continuous data, although its use is most often with continuous data see our Types of Variable guide for data types. The mean is equal to the sum of all the values in the data set divided by the number of values in the data set. You may have noticed that the above formula refers to the sample mean. So, why have we called it a sample mean?
This is because, in statistics, samples and populations have very different meanings and these differences are very important, even if, in the case of the mean, they are calculated in the same way. The mean is essentially a model of your data set. It is the value that is most common. You will notice, however, that the mean is not often one of the actual values that you have observed in your data set.
However, one of its important properties is that it minimises error in the prediction of any one value in your data set. That is, it is the value that produces the lowest amount of error from all other values in the data set.
An important property of the mean is that it includes every value in your data set as part of the calculation. In addition, the mean is the only measure of central tendency where the sum of the deviations of each value from the mean is always zero. The mean has one main disadvantage: it is particularly susceptible to the influence of outliers. These are values that are unusual compared to the rest of the data set by being especially small or large in numerical value.
For example, consider the wages of staff at a factory below:. Staff 1 2 3 4 5 6 7 8 9 10 Salary 15k 18k 16k 14k 15k 15k 12k 17k 90k 95k. The mean is being skewed by the two large salaries. Therefore, in this situation, we would like to have a better measure of central tendency. As we will find out later, taking the median would be a better measure of central tendency in this situation.
Another time when we usually prefer the median over the mean or mode is when our data is skewed i. If we consider the normal distribution - as this is the most frequently assessed in statistics - when the data is perfectly normal, the mean, median and mode are identical.
Moreover, they all represent the most typical value in the data set. Center of a data set: mean, median, mode Back to Course Index. Don't just watch, practice makes perfect.
A school collected gifts from students to give to needy children in the community over the Christmas holidays. The following numbers of gifts were collected. Grade Total number of gifts collected 1 65 2 3 70 4 54 5 45 6 54 7 What are the median and mean?
Round your answer to the nearest whole number. Which value is an outlier? Which measure of central tendency better describes the data? Explain why. The following table represents the sizes of winter boots that were sold last week. What are the mean and mode sizes of shoes sold? Round your mean to the nearest whole shoe size. If you were in charge of ordering in more boots, which measure of central tendency is more meaningful? In a running race, the mean time was 2 minutes; the mode time was 2 minutes and 10 seconds; and the median time was 1 minute and 55 seconds.
Jack had a time of 2 minutes. Homoscedasticity, or homogeneity of variances, is an assumption of equal or similar variances in different groups being compared. This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. Uneven variances in samples result in biased and skewed test results.
Statistical tests such as variance tests or the analysis of variance ANOVA use sample variance to assess group differences of populations. They use the variances of the samples to assess whether the populations they come from significantly differ from each other. Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. Both measures reflect variability in a distribution, but their units differ:.
Although the units of variance are harder to intuitively understand, variance is important in statistical tests. The empirical rule, or the In a normal distribution , data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. The measures of central tendency mean, mode and median are exactly the same in a normal distribution. The median is the most informative measure of central tendency for skewed distributions or distributions with outliers.
For example, the median is often used as a measure of central tendency for income distributions, which are generally highly skewed. In contrast, the mean and mode can vary in skewed distributions. Because the range formula subtracts the lowest number from the highest number, the range is always zero or a positive number. In statistics, the range is the spread of your data from the lowest to the highest value in the distribution.
It is the simplest measure of variability. While central tendency tells you where most of your data points lie, variability summarizes how far apart your points from each other. Data sets can have the same central tendency but different levels of variability or vice versa.
Together, they give you a complete picture of your data. Variability is most commonly measured with the following descriptive statistics :. Variability tells you how far apart points lie from each other and from the center of a distribution or a data set. While interval and ratio data can both be categorized, ranked, and have equal spacing between adjacent values, only ratio scales have a true zero. For example, temperature in Celsius or Fahrenheit is at an interval scale because zero is not the lowest possible temperature.
In the Kelvin scale, a ratio scale, zero represents a total lack of thermal energy. A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test.
It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data i.
The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution a. In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.
A t -score a. The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution.
The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. It is a type of normal distribution used for smaller sample sizes, where the variance in the data is unknown.
The t -distribution forms a bell curve when plotted on a graph. It can be described mathematically using the mean and the standard deviation. The correlation coefficient only tells you how closely your data fit on a line, so two datasets with the same correlation coefficient can have very different slopes. Correlation coefficients always range between -1 and 1. The sign of the coefficient tells you the direction of the relationship: a positive value means the variables change together in the same direction, while a negative value means they change together in opposite directions.
The absolute value of a number is equal to the number without its sign. The absolute value of a correlation coefficient tells you the magnitude of the correlation: the greater the absolute value, the stronger the correlation. A correlation coefficient is a single number that describes the strength and direction of the relationship between your variables.
Different types of correlation coefficients might be appropriate for your data based on their levels of measurement and distributions. A power analysis is a calculation that helps you determine a minimum sample size for your study. If you know or have estimates for any three of these, you can calculate the fourth component.
In statistical hypothesis testing , the null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship. Statistical analysis is the main method for analyzing quantitative research data. It uses probabilities and models to test predictions about a population from sample data.
The risk of making a Type II error is inversely related to the statistical power of a test. Power is the extent to which a test can correctly detect a real effect when there is one. To indirectly reduce the risk of a Type II error, you can increase the sample size or the significance level to increase statistical power. The risk of making a Type I error is the significance level or alpha that you choose. The significance level is usually set at 0. In statistics, ordinal and nominal variables are both considered categorical variables.
Even though ordinal data can sometimes be numerical, not all mathematical operations can be performed on them. In statistics, power refers to the likelihood of a hypothesis test detecting a true effect if there is one. A statistically powerful test is more likely to reject a false negative a Type II error. Your study might not have the ability to answer your research question. While statistical significance shows that an effect exists in a study, practical significance shows that the effect is large enough to be meaningful in the real world.
Statistical significance is denoted by p -values whereas practical significance is represented by effect sizes. There are dozens of measures of effect sizes. Effect size tells you how meaningful the relationship between variables or the difference between groups is.
A large effect size means that a research finding has practical significance, while a small effect size indicates limited practical applications. Using descriptive and inferential statistics , you can make two types of estimates about the population : point estimates and interval estimates. Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie. Standard error and standard deviation are both measures of variability.
The standard deviation reflects variability within a sample, while the standard error estimates the variability across samples of a population. The standard error of the mean , or simply standard error , indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.
To figure out whether a given number is a parameter or a statistic , ask yourself the following:. If the answer is yes to both questions, the number is likely to be a parameter. For small populations, data can be collected from the whole population and summarized in parameters. If the answer is no to either of the questions, then the number is more likely to be a statistic.
The arithmetic mean is the most commonly used mean. But there are some other types of means you can calculate depending on your research purposes:. You can find the mean , or average, of a data set in two simple steps:. This method is the same whether you are dealing with sample or population data or positive or negative numbers.
Multiple linear regression is a regression model that estimates the relationship between a quantitative dependent variable and two or more independent variables using a straight line. The 3 main types of descriptive statistics concern the frequency distribution, central tendency, and variability of a dataset. Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.
In statistics, model selection is a process researchers use to compare the relative value of different statistical models and determine which one is the best fit for the observed data. The Akaike information criterion is one of the most common methods of model selection. AIC weights the ability of the model to predict the observed data against the number of parameters the model requires to reach that level of precision. AIC model selection can help researchers find a model that explains the observed variation in their data while avoiding overfitting.
In statistics, a model is the collection of one or more independent variables and their predicted interactions that researchers use to try to explain variation in their dependent variable. You can test a model using a statistical test. The Akaike information criterion is calculated from the maximum log-likelihood of the model and the number of parameters K used to reach that likelihood. The AIC function is 2K — 2 log-likelihood.
Lower AIC values indicate a better-fit model, and a model with a delta-AIC the difference between the two AIC values being compared of more than -2 is considered significantly better than the model it is being compared to.
The Akaike information criterion is a mathematical test used to evaluate how well a model fits the data it is meant to describe. It penalizes models which use more independent variables parameters as a way to avoid over-fitting. AIC is most often used to compare the relative goodness-of-fit among different models under consideration and to then choose the model that best fits the data. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant result.
Significant differences among group means are calculated using the F statistic, which is the ratio of the mean sum of squares the variance explained by the independent variable to the mean square error the variance left over. If the F statistic is higher than the critical value the value of F that corresponds with your alpha value, usually 0. If you are only testing for a difference between two groups, use a t-test instead. The formula for the test statistic depends on the statistical test being used.
Generally, the test statistic is calculated as the pattern in your data i. Linear regression most often uses mean-square error MSE to calculate the error of the model. MSE is calculated by:. Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE. Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line.
Both variables should be quantitative. For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands.
This linear relationship is so certain that we can use mercury thermometers to measure temperature. A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line or a plane in the case of two or more independent variables.
A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary. A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.
A one-sample t-test is used to compare a single population to a standard value for example, to determine whether the average lifespan of a specific town is different from the country average.
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