Descriptive statistics

# Descriptive statistics

Discussion

Encyclopedia
Descriptive statistics quantitatively describe the main features of a collection of data
Data
The term data refers to qualitative or quantitative attributes of a variable or set of variables. Data are typically the results of measurements and can be the basis of graphs, images, or observations of a set of variables. Data are often viewed as the lowest level of abstraction from which...

. Descriptive statistics are distinguished from inferential statistics
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...

(or inductive statistics), in that descriptive statistics aim to summarize a data set, rather than use the data to learn about the population
Statistical population
A statistical population is a set of entities concerning which statistical inferences are to be drawn, often based on a random sample taken from the population. For example, if we were interested in generalizations about crows, then we would describe the set of crows that is of interest...

that the data are thought to represent. This generally means that descriptive statistics, unlike inferential statistics, are not developed on the basis of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

theory. Even when a data analysis draws its main conclusions using inferential statistics, descriptive statistics are generally also presented. For example in a paper reporting on a study involving human subjects, there typically appears a table giving the overall sample size
Sample size
Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...

, sample sizes in important subgroups (e.g., for each treatment or exposure group), and demographic or clinical characteristics such as the average age, the proportion of subjects of each sex, and the proportion of subjects with related comorbidities
Comorbidity
In medicine, comorbidity is either the presence of one or more disorders in addition to a primary disease or disorder, or the effect of such additional disorders or diseases.- In medicine :...

.

## Use in statistical analyses

Descriptive statistics provide simple summaries about the sample and the measures. Together with simple graphics analysis, they form the basis of quantitative analysis of data.

Descriptive statistics summarize data. For example, the shooting percentage
Percentage
In mathematics, a percentage is a way of expressing a number as a fraction of 100 . It is often denoted using the percent sign, “%”, or the abbreviation “pct”. For example, 45% is equal to 45/100, or 0.45.Percentages are used to express how large/small one quantity is, relative to another quantity...

Basketball is a team sport in which two teams of five players try to score points by throwing or "shooting" a ball through the top of a basketball hoop while following a set of rules...

is a descriptive statistic that summarizes the performance of a player or a team. This number is the number of shots made divided by the number of shots taken. A player who shoots 33% is making approximately one shot in every three. One making 25% is hitting once in four. The percentage summarizes or describes multiple discrete events. Or, consider the scourge of many students, the grade point average. This single number describes the general performance of a student across the range of their course experiences.

Describing a large set of observations with a single indicator risks distorting the original data or losing important detail. For example, the shooting percentage doesn't tell you whether the shots are three-pointers or lay-ups, and GPA doesn't tell you whether the student was in difficult or easy courses. Despite these limitations, descriptive statistics provide a powerful summary that may enable comparisons across people or other units.

### Univariate analysis

Univariate analysis
Univariate analysis
Univariate analysis is the simplest form of quantitative analysis. The analysis is carried out with the description of a single variable and its attributes of the applicable unit of analysis...

involves the examination across cases of a single variable, focusing on three characteristics: the distribution; the central tendency; and the dispersion. It is common to compute all three for each study variable.

#### Distribution

The distribution is a summary of the frequency of individual or ranges of values for a variable. The simplest distribution would list every value of a variable and the number of cases who had that value. For instance, computing the distribution of gender in the study population means computing the percentages that are male and female. The gender variable has only two, making it possible and meaningful to list each one. However, this does not work for a variable such as income that has many possible values. Typically, specific values are not particularly meaningful (income of 50,000 is typically not meaningfully different from 51,000). Grouping the raw scores using ranges of values reduces the number of categories to something more meaningful. For instance, we might group incomes into ranges of 0-10,000, 10,001-30,000, etc.

Frequency distributions are depicted as a table or as a graph. Table 1 shows an age frequency distribution with five categories of age ranges defined. The same frequency distribution can be depicted in a graph as shown in Figure 2. This type of graph is often referred to as a histogram
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...

or bar chart.

#### Central tendency

The central tendency
Central tendency
In statistics, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value...

of a distribution locates the "center" of a distribution of values. The three major types of estimates of central tendency are the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

, the median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...

, and the mode
Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....

.

The mean is the most commonly used method of describing central tendency. To compute the mean, take the sum of the values and divide by the count. For example, the mean quiz score is determined by summing all the scores and dividing by the number of students taking the exam. For example, consider the test score values:

15, 20, 21, 36, 15, 25, 15

The sum of these 7 values is 147, so the mean is 147/7 =21.

The median is the score found at the middle of the set of values, i.e., that has as many cases with a larger value as have a smaller value. One way to compute the median is to sort the values in numerical order, and then locate the value in the middle of the list. For example, if there are 500 values, the median is the average of the two values in 250th and 251st positions. If there are 499 values, the value in 250th position is the median. Sorting the 7 scores above produces:

15, 15, 15, 20, 21, 25, 36

There are 7 scores and score #4 represents the halfway point. The median is 20. If there are an even number of observations, then the median is the mean of the two middle scores. In the example, if there were an 8th observation, with a value of 25, the median becomes the average of the 4th and 5th scores, in this case 20.5.

The mode is the most frequently occurring value in the set. To determine the mode, compute the distribution as above. The mode is the value with the greatest frequency. In the example, the modal value 15, occurs three times. In some distributions there is a "tie" for the highest frequency, i.e., there are multiple modal values. These are called multi-modal distributions.

Notice that the three measures typically produce different results. The term "average" obscures the difference between them and is better avoided.
The three values are equal if the distribution is perfectly "normal" (i.e., bell-shaped).

#### Dispersion

Dispersion is the spread of values around the central tendency. There are two common measures of dispersion, the range
Range (statistics)
In the descriptive statistics, the range is the length of the smallest interval which contains all the data. It is calculated by subtracting the smallest observation from the greatest and provides an indication of statistical dispersion.It is measured in the same units as the data...

and the standard deviation
Standard deviation
Standard deviation is a widely used measure of variability or diversity used in statistics and probability theory. It shows how much variation or "dispersion" there is from the average...

. The range is simply the highest value minus the lowest value. In our example distribution, the high value is 36 and the low is 15, so the range is 36 − 15 = 21.

The standard deviation is a more accurate and detailed estimate of dispersion because an outlier can greatly exaggerate the range (as was true in this example where the single outlier value of 36 stands apart from the rest of the values). The standard deviation shows the relation that set of scores has to the mean of the sample. Again let's take the set of scores:

15, 20, 21, 36, 15, 25, 15

to compute the standard deviation, we first find the distance between each value and the mean. We know from above that the mean is 21. So, the differences from the mean are:
15 − 21 = −6
20 − 21 = −1
21 − 21 = 0
36 − 21 = 15
15 − 21 = −6
25 − 21 = +4
15 − 21 = −6

Notice that values that are below the mean have negative differences and values above it have positive ones. Next, we square each difference:
(−6)2 = 36
(−1)2 = 1
(+0)2 = 0
(15)2 = 225
(−6)2 = 36
(+4)2 = 16
(−6)2 = 36

Now, we take these "squares" and sum them to get the sum of squares (SS) value. Here, the sum is 350. Next, we divide this sum by the number of scores minus 1. Here, the result is 350 / 6 = 58.3. This value is known as the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

. To get the standard deviation, we take the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of the variance (remember that we squared the deviations earlier). This would be √58.3 = 7.63.

Although this computation may seem convoluted, it's actually quite simple. In English, we can describe the standard deviation as:

"the square root of the sum of the squared deviations from the mean divided by the number of scores minus one"

The standard deviation allows us to reach some conclusions about specific scores in our distribution. Assuming that the distribution of scores is close to "normal", the following conclusions can be reached:
• approximately 68% of the scores in the sample fall within one standard deviation of the mean
• approximately 95% of the scores in the sample fall within two standard deviations of the mean
• approximately 99% of the scores in the sample fall within three standard deviations of the mean

For instance, since the mean in our example is 21 and the standard deviation is 7.63, we can from the above statement estimate that approximately 95% of the scores will fall in the range of 21 − (2×7.63) to 21 + (2×7.63) or between 5.74 and 36.26. Values beyond two standard deviations from the mean can be considered "outlier
Outlier
In statistics, an outlier is an observation that is numerically distant from the rest of the data. Grubbs defined an outlier as: An outlying observation, or outlier, is one that appears to deviate markedly from other members of the sample in which it occurs....

s". 36 is the only such value in our distribution. Outliers help identify observations for further analysis or possible problems in the observations. Standard deviations also convert measures on very different scales, such as height and weight, into values that can be compared.

#### Other statistics

In research involving comparisons between groups, emphasis is often placed on the significance level
Statistical significance
In statistics, a result is called statistically significant if it is unlikely to have occurred by chance. The phrase test of significance was coined by Ronald Fisher....

for the hypothesis that the groups being compared differ to a degree greater than would be expected by chance. This significance level is often represented as a p-value
P-value
In statistical significance testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. One often "rejects the null hypothesis" when the p-value is less than the significance level α ,...

, or sometimes as the standard score
Standard score
In statistics, a standard score indicates how many standard deviations an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation...

of a test statistic
Test statistic
In statistical hypothesis testing, a hypothesis test is typically specified in terms of a test statistic, which is a function of the sample; it is considered as a numerical summary of a set of data that...

. In contrast, an effect size
Effect size
In statistics, an effect size is a measure of the strength of the relationship between two variables in a statistical population, or a sample-based estimate of that quantity...

conveys the estimated magnitude and direction of the difference between groups, without regard to whether the difference is statistically significant. Reporting significance levels without effect sizes is problematic, since for large sample sizes even small effects of little practical importance can be statistically significant.

## Examples of descriptive statistics

Most statistics can be used either as a descriptive statistic, or in an inductive analysis. For example, we can report the average reading test score for the students in each classroom in a school, to give a descriptive sense of the typical scores and their variation. If we perform a formal hypothesis test on the scores, we are doing inductive rather than descriptive analysis.

Some statistical summaries are especially common in descriptive analyses. Some examples follow.
• Measures of central tendency
Central tendency
In statistics, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value...

• Measures of dispersion
Statistical dispersion
In statistics, statistical dispersion is variability or spread in a variable or a probability distribution...

• Measures of association
Association (statistics)
In statistics, an association is any relationship between two measured quantities that renders them statistically dependent. The term "association" refers broadly to any such relationship, whereas the narrower term "correlation" refers to a linear relationship between two quantities.There are many...

• Cross-tabulation, contingency table
Contingency table
In statistics, a contingency table is a type of table in a matrix format that displays the frequency distribution of the variables...

• Histogram
Histogram
In statistics, a histogram is a graphical representation showing a visual impression of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson...

• Quantile
Quantile
Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. Dividing ordered data into q essentially equal-sized data subsets is the motivation for q-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets...

, Q-Q plot
Q-Q plot
In statistics, a Q-Q plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other. First, the set of intervals for the quantiles are chosen...

• Scatterplot
Scatterplot
A scatter plot or scattergraph is a type of mathematical diagram using Cartesian coordinates to display values for two variables for a set of data....

• Box plot
Box plot
In descriptive statistics, a box plot or boxplot is a convenient way of graphically depicting groups of numerical data through their five-number summaries: the smallest observation , lower quartile , median , upper quartile , and largest observation...

• Summary statistics
Summary statistics
In descriptive statistics, summary statistics are used to summarize a set of observations, in order to communicate the largest amount as simply as possible...

• Exploratory data analysis
Exploratory data analysis
In statistics, exploratory data analysis is an approach to analysing data sets to summarize their main characteristics in easy-to-understand form, often with visual graphs, without using a statistical model or having formulated a hypothesis...

• Statistical inference
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...

• Data mining
Data mining
Data mining , a relatively young and interdisciplinary field of computer science is the process of discovering new patterns from large data sets involving methods at the intersection of artificial intelligence, machine learning, statistics and database systems...