Introduction to Descriptive Statistics
- 1. Introduction to Descriptive Statistics I Sanju Rusara Seneviratne MBPsS
- 2. Overview of Intro to Descriptive Statistics I This lecture will cover the following topics: Definition and Types of Descriptive Statistics Mean, Median, Mode and Range Skewness and Kurtosis Normality Curve Variance and Standard Deviation Quartiles Percentiles Using Excel for Descriptive Statistics
- 3. Defining Descriptive Statistics The analysis of data that helps describe, show or summarize data in a meaningful way such that, for example, patterns might emerge from the data. They do not, however, allow us to make conclusions beyond the data we have analyzed or reach conclusions regarding any hypotheses we might have made. Descriptive vs. Inferential: Descriptive statistics are used to describe our samples and inferential statistics are used to generalize from our samples to the wider population.
- 4. Types of Descriptive Statistic 1. Measures of central tendency: These are ways of describing the central position of a frequency distribution for a group of data. We can describe this central position using a number of statistics, including the mode, median, and mean. 2. Measures of spread: These are ways of summarizing a group of data by describing how spread out the scores are. Measures of spread help us to summarize how spread out data are. To describe this spread, a number of statistics are available us, including the range, quartiles, absolute deviation, variance and standard deviation.
- 5. Summarizing Descriptive Statistics When we use descriptive statistics it is useful to summarize our group of data using a combination of: • tabulated description (i.e., tables) • graphical description (i.e., graphs and charts) • statistical commentary (i.e., a discussion of the results)
- 6. Mean, Median, Mode and Range • Mean - The mean is the average of all numbers and is sometimes called the arithmetic mean. To calculate mean, add all of the in a set and then divide the sum by the total count of numbers. • Median - The statistical median is the middle number in a sequence of numbers. To find the median, organize each number in order by size; the number in the middle is the median. • Mode - The mode is the number that occurs most often within a set of numbers. • Range - The range is the difference between the highest and lowest values within a set of numbers. To calculate range, subtract the smallest number from the largest number in the set.
- 7. Skewness and Kurtosis • Skewness - a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point. • Kurtosis - a measure of whether the data are heavy- tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light or lack of outliers. A uniform distribution would be the extreme case. • The histogram is an effective graphical technique for showing both the skewness and kurtosis of data set.
- 8. Normality Curve • The normal distribution is the most important and most widely used distribution in statistics. It is sometimes called the "bell curve” and the "Gaussian curve”.
- 9. Seven Features of Normal Distributions 1. Normal distributions are symmetric around their mean. 2. The mean, median, and mode of a normal distribution are equal. 3. The area under the normal curve is equal to 1.0. 4. Normal distributions are denser in the center and less dense in the tails. 5. Normal distributions are defined by two parameters, the mean (μ) and the standard deviation (σ). 6. 68% of the area of a normal distribution is within one standard deviation of the mean. 7. Approximately 95% of the area of a normal distribution is within two standard deviations of the mean.
- 10. Variance and Standard Deviation • Variance: measures how far a data set is spread out. The technical definition is “The average of the squared differences from the mean,” but all it really does is to give you a very general idea of the spread of your data. A value of zero means that there is no variability; All the numbers in the data set are the same. • Standard Deviation: the square root of the variance. While variance gives you a rough idea of spread, the standard deviation is more concrete, giving you exact distances from the mean.
- 11. Quartiles • Quartiles in statistics are values that divide your data into quarters. They divide your data into four segments according to where the numbers fall on the number line. • The four quarters that divide a data set into quartiles are: The lowest 25% of numbers. The next lowest 25% of numbers (up to the median). The second highest 25% of numbers (above the median). The highest 25% of numbers.
- 12. Percentiles • The most common definition of a percentile is a number where a certain percentage of scores fall below that number. The 25th percentile is also called the first quartile. The 50th percentile is generally the median (if you’re using the third definition— see below). The 75th percentile is also called the third quartile. The difference between the third and first quartiles is the interquartile range. • Percentile Rank: The nth percentile is the lowest score that is greater than a certain percentage (“n”) of the scores. The nth percentile is the smallest score that is greater than or equal to a certain percentage of the scores. To rephrase this, it’s the percentage of data that falls at or below a certain observation. • A percentile range is the difference between two specified percentiles.
- 13. Conducting Descriptive Analysis in Excel • Step 1: Type your data into Excel, in a single column. For example, if you have ten items in your data set, type them into cells A1 through A10. • Step 2: Click the “Data” tab and then click “Data Analysis” in the Analysis group. • Step 3: Highlight “Descriptive Statistics” in the pop-up Data Analysis window. • Step 4: Type an input range into the “Input Range” text box. For this example, type “A1:A10” into the box.
- 14. Conducting Descriptive Analysis in Excel • Step 5: Check the “Labels in first row” check box if you have titled the column in row 1, otherwise leave the box unchecked. • Step 6: Type a cell location into the “Output Range” box. For example, type “C1.” Make sure that two adjacent columns do not have data in them. • Step 7: Click the “Summary Statistics” check box and then click “OK” to display Excel descriptive statistics. A of descriptive statistics will be returned in the column you selected as the Output Range.
- 15. Introduction to Descriptive Statistics II Sanju Rusara Seneviratne MBPsS
- 16. Overview of Intro to Descriptive Statistics II This lecture will cover the following topics: Bar Charts Pie Charts Histograms Box-Plots Scatter Plots
- 17. Bar Charts • A bar graph (also known as a bar chart or bar diagram) is a visual tool that uses bars to compare data among categories. A bar graph may run horizontally or vertically. The important thing to know is that the longer the bar, the greater its value. • Bar graphs consist of two axes. On a vertical bar graph, the horizontal axis (or x-axis) shows the data categories. The vertical axis (or y-axis) is the scale.
- 18. Bar Charts • Bar graphs have three key attributes: 1. A bar diagram makes it easy to compare sets of data between different groups at a glance. 2. The graph represents categories on one axis and a discrete value in the other. The goal is to show the relationship between the two axes. 3. Bar charts can also show big changes in data over time.
- 19. Examples of Bar Charts
- 20. Examples of Bar Charts
- 21. Pie Charts • A pie chart is a circular graph that shows the relative contribution that different categories contribute to an overall total. • A wedge of the circle represents each category’s contribution, such that the graph resembles a pie that has been cut into different sized slices. • Every 1% contribution that a category contributes to the total corresponds to a slice with an angle of 3.6 degrees.
- 22. Pie Charts • Pie charts are a visual way of displaying data that might otherwise be given in a small table. • Pie charts are useful for displaying data that are classified into nominal or ordinal categories. Nominal data are categorised according to descriptive or qualitative information such as county of birth or type of pet owned. Ordinal data are similar but the different categories can also be ranked, for example in a survey people may be asked to say whether they classed something as very poor, poor, fair, good, very good.
- 23. Pie Charts • Pie charts are generally used to show percentage or proportional data and usually the percentage represented by each category is provided next to the corresponding slice of pie. • Pie charts are good for displaying data for around 6 categories or fewer. When there are more categories it is difficult for the eye to distinguish between the relative sizes of the different sectors and so the chart becomes difficult to interpret.
- 24. Examples of Pie Charts
- 25. Examples of Pie Charts
- 26. Histograms • A histogram is a plot that lets you discover, and show, the underlying frequency distribution (shape) of a set of continuous data. This allows the inspection of the data for its underlying distribution (e.g., normal distribution), outliers, skewness, etc. • The area of the bar that indicates the frequency of occurrences for each bin. This means that the height of the bar does not necessarily indicate how many occurrences of scores there were within each individual bin. It is the product of height multiplied by the width of the bin that indicates the frequency of occurrences within that bin.
- 27. Histograms • One of the reasons that the height of the bars is often incorrectly assessed as indicating frequency and not the area of the bar is due to the fact that a lot of histograms often have equally spaced bars (bins), and under these circumstances, the height of the bin does reflect the frequency. • The major difference is that a histogram is only used to plot the frequency of score occurrences in a continuous data set that has been divided into classes, called bins. Bar charts, on the other hand, can be used for a great deal of other types of variables including ordinal and nominal data sets.
- 28. Histograms A histogram showing frequencies of different age groups in a sample. Thinking Point: What can you infer about the normal distribution of this data from this chart?
- 29. Box-Plots • A boxplot is a standardized way of displaying the distribution of data based on a five number summary (“minimum”, first quartile (Q1), median, third quartile (Q3), and “maximum”). • It can tell you about your outliers and what their values are. • It can also tell you if your data is symmetrical, how tightly your data is grouped, and if and how your data is skewed.
- 30. Example of a Box-Plot See next slide for description of this box-plot.
- 31. Elements of a Box-Plot • A boxplot is a graph that gives you a good indication of how the values in the data are spread out. median (Q2/50th Percentile): the middle value of the dataset. first quartile (Q1/25th Percentile): the middle number between the smallest number (not the “minimum”) and the median of the dataset. third quartile (Q3/75th Percentile): the middle value between median and the highest value (not the “maximum”) of the dataset. interquartile range (IQR): 25th to the 75th percentile. whiskers (shown in blue) outliers (shown as green circles) “maximum”: Q3 + 1.5*IQR “minimum”: Q1 -1.5*IQR
- 32. Scatter Plots • A scatter plot is a two-dimensional data visualization that uses dots to represent the values obtained for two different variables - one plotted along the x-axis and the other plotted along the y-axis. • Scatter plots are used when you want to show the relationship between two variables. Scatter plots are sometimes called correlation plots because they show how two variables are correlated. • However, not all relationships are linear.
- 33. Examples of Scatter Plots A scatterplot showing the relationship between weight (in lb) and height (in inches) in children. This demonstrates a positive linear relationship.
- 34. Examples of Scatter Plots
- 35. References and Further Reading Books: • Dancey, C. and Reidy, J. (2017). Statistics without Maths for Psychology,7th Edition. New York: Pearson. • Howitt, D., & Cramer, D. (2017). Statistics in psychology using SPSS. New York: Pearson. Articles: • Bickel, P. J., & Lehmann, E. L. (1975). Descriptive Statistics for Nonparametric Models I. Introduction. The Annals of Statistics, 3(5), 1038-1044. doi:10.1214/aos/1176343239 | https://link.springer.com/content/pdf/10.1007/978-1- 4614-1412-4_42.pdf