Measures of Central Tendency

In statistics, the central tendency refers to a single value of a dataset or a whole distribution. The central tendency is a typical value of the dataset. Central tendency of a dataset can be identified using measures such as mode, median, and mean (see Figure 10.1).

Measures of Dispersion

The measure of dispersion, as the name indicates, describes the scattering of data. It explains the variation in data points and gives a clear view of the data distribution. It indicates the heterogeneity or homogeneity of distributed data observations, which are categorized as the absolute measure and relative measure of dispersion:

• The absolute measure of dispersion is used for observation scattering in distances such as quartile deviation and range. It denotes differences in observations for the average number of deviations such as standard and mean deviations.

• A relative measure of dispersion compares the distributed data with two or more data points.

These include the coefficient of mean deviation, coefficient of range, coefficient of variation, coefficient of quartile deviation, and coefficient of standard deviation.

Range

Range is an easily understandable measure of dispersion. Range is the difference between the maximum value and the minimum value of a dataset. If Xi_min and Xi_max are these two values, the range can be identified using this formula:

Range = Xi_max – Xi_min

Quartile Deviation

Quartiles divide a set of data into quarters. Following are the specifics:

• The middle number between the median of the dataset and the smallest number is the first quartile (Qu₁).

• The dataset median is the second quartile (Qu₂).

• The middle number between the largest number and the median is the third quartile (Qu₃).

The formula for quartile deviation is as follows:

Quartile deviation = (Qu₃ - Qu₁)/2

Quartile deviation is the best dispersion measure for open-ended classification. It is independent of origin change and dependent on scale change.

Frequencies

Frequency (f) refers to the number of times an observation of a specific value occurs in data. For example, frequency is the number of times that each variable occurs, such as the number of male athletes and the number of female athletes within a sample population.

A distribution represents frequency pattern of a variable. A distribution is a set of probable values as well as their frequencies. In other words, a frequency distribution represents values and their frequency—that is, how often each value occurs in a sample dataset.

Let’s consider an example of a frequency distribution of magazines sold at a retail outlet. The data includes the numbers of magazines sold at a local retail outlet over the past 7 days:

10, 11, 12, 13, 14, 15, 16

Table 10.4 shows how many times each number occurs in this dataset.

As you can see, the frequency tells how often there were 10, 11, 12, 13, 14, 15, or 16 magazines sold over the past 7 days.

A distribution of frequency could indicate either a real number of values falling in observation percentage or a range. In the observation percentage, the distribution of frequency is referred to as a relative distribution of frequency. Distributions of frequency tables are leveraged for numeric variables and categorical variables. Continuous variables in a frequency distribution are used only with class intervals.

Percent Change and Percent Difference

Percent change can be used to compare old values with new values. Estimating the percent change between two provided quantities is an effortless process. When an old or initial value and a new or final value of quantity are identified, the percent change formula is applied to identify the percent change. The formula is as follows:

$\text{Percent change}=\frac{(V_2-V_1)}{\left|V_1\right|}\times 100$

where:

V₂ denotes the new value

V₁ denotes the old value

If the percent change value is positive, this indicates that the percentage has increased; if the percent change value is negative, this indicates that the percentage has decreased.

For example, if V₁ = 100 and V₂ = 200, the percent change would be:

$\begin{array}{lll}\text{Percent change}& =& \left[(200-100)/100\right]\times 100\\ & =& [100/100]\times 100\\ & =& 100\%\end{array}$

Percent difference is a measure of the absolute value of variance among the two numbers, divided by the average of the two values and then multiplied by 100%:

$\begin{array}{c}\text{Percent difference}=\frac{|V_1-V_2|}{\left[\frac{(V_2-V_1)}{2}\right]}\times 100\end{array}$

Again using the example of V₁ = 100 and V₂ = 200:

$\begin{array}{lll}\text{Percent difference}& =& |100-200|/\left[(100+200)/2\right]\times 100\\ & =& |-100|/\left[150\right]\times 100\\ & =& 0.66667\times 100\\ & =& 66.667\end{array}$

Confidence Intervals

The confidence level is the range of values that are likely to contain plausible population values. Confidence intervals are used to measure the degree of uncertainty in a sampling method. Typically, the most common confidence interval is 95% or 99%; however, it is possible to have other values, such as 85% or 90%.

For example, if a data analyst constructs a confidence interval with a 95% confidence level, that analyst is confident that 95 out of 100 times, the estimate will fall between the upper and lower values specified by the confidence interval (see Figure 10.2).

Let’s consider another example. Say that a device manufacturer wants to ensure that the weight of the devices that will be carried on a military convoy is in line with specifications. A data analyst has measured the average weight of a sample of 100 devices to be 10 kg. He has also found the 95% confidence interval to be between 9.6 kg and 10.30 kg. This means the data analyst can be 95% sure that the average weight of all the devices manufactured will be between 9.6 kg and 10.30 kg.

The formula for the confidence interval is:

$\mathbf{\text{CI=}}\overline{\mathbf{\text{X}}}\mathbf{\pm }\mathbf{\text{Z}}\mathbf{\times }\frac{\mathbf{\sigma }}{\sqrt{\mathbf{n}}}$

where:

X is the sample mean

Z is the confidence coefficient (Z-score), which is 1.960 for 95% and 2.576 for 99%

σ is the standard deviation

n is the sample size

Z-score

A Z-score, also referred to as a standard score, describes how distant from the mean a data point is. It is also an estimation of how many standard deviations above or below the mean of a population a raw score is.

The basic formula for the Z-score is:

Z = (x – µ) / σ

where:

x denotes the observed value

σ denotes the standard deviation of the sample

µ denotes the mean of the sample

If the x value is 1350, the mean value is 1000, and the standard deviation value is 200, then the Z-value is found as follows:

$\begin{array}{ll}Z=& (1350-1000)/200\\ =& 1.750\end{array}$

In this case, the Z-score is 1.750 standard deviations above the mean.

Figure 10.3 illustrates a sample Z-score value.

How is a Z-score helpful? A Z-score gives you an idea about how an individual value compares to the rest of the distribution. For example, a Z-score can tell you if devices being produced for military vehicles are being actively used or not, depending upon the weight specifications on the vehicles they are supposed to be used with.

In the previous example, we have a positive Z-score (that is, the individual value is greater than the mean, as shown in Figure 10.3). A negative Z-score indicates a value less than the mean. Finally, a Z-score of 0 means that the individual value is equal to the mean.

t-tests

The t-test, which originated in inferential statistics, is used to identify the main variance between mean values from two groups. In other words, it is used to compare the mean values from two samples and evaluate whether the means of the two groups are statistically dissimilar. t-tests are based on hypotheses.

There are three categories of t-tests:

• One-sample t-test: This type of t-test compares the average or mean of one group against the set average or population mean. For example, it can be used to compare the sales of a product across a set of new stores against sales in one of the established stores.

• Two-sample t-test: This type of t-test is used to compare the means of two different samples and implies that the two mean groups are different from the hypothesized population mean. For example, it might be used to compare the performance of salespeople across two different states for the same product.

• Paired t-test: This type of t-test is used to compare separate means for a group at two different times or under two different conditions. For example, it might be used to compare the effect of training of salespeople on selling a sophisticated product before and after training.

Figure 10.4 shows the distribution of t-values when the null hypothesis is true.

p-values

A p-value, or probability value, is leveraged as part of hypothesis testing to help accept or reject the null hypothesis. The p-value is a number that explains the probability that a data point happened by chance (that is, randomly). The statistical significance level is denoted as a p-value between 0 and 1.

p-values are usually expressed as decimal figures and can be expressed as percentages. For example, a p-value of 0.054 is 5.40% and implies that there is a 5.40% chance that the results could be random (or happened by chance). In comparison, a p-value of 0.99 translates to 99.00% and implies that the results have a 99% probability of being completely random. Hence, with p-values, a smaller value reflects more significant results. Table 10.5 explains how to interpret p-values.

Chi-Square Test

The chi-square test is used for testing hypotheses about observed distributions in various categories. In other words, a chi-square test compares the observed values in a dataset to the expected values that you would see if the null hypothesis were true. A chi-square test can be used to infer information such as:

• Whether two categorical variables are independent and have no relationship with one another. This is known as a chi-square test of independence.

• Whether one variable follows a given hypothesized distribution or not. This is known as a chi-square goodness-of-fit test.

Chi-square can be calculated using the following formula:

X_c² = Σ(Ob_i – E_i)² / E_i

where:

c denotes the degrees of freedom

E represents the expected value

Ob represents the observed value

A chi-square test provides the p-value, which indicates whether the results of the test are significant (as discussed in the previous section).

Let’s consider an example. An organization is conducting research and trying to relate the different levels of education of people to whether those people work in IT or non-IT jobs. Table 10.6 shows the simple random sample the organization is working with.

The organization can use a chi-square test of independence to determine whether there is a statistically significant association between the two variables (education and working in IT jobs).

In this case, the chi-square (X²) test gives the number 20.507268, with a p-value of 0.000035. Now, given that the p-value is less than 0.05, we know that result of the chi-square test is statistically significant. Thus, the alternative hypothesis is acceptable (and rejects the null hypothesis), and there is sufficient evidence to state that there is an association between education level and holding an IT job.

The graph in Figure 10.5 shows the chi-squared distribution graph for this example.

Hypothesis Testing

We began looking at hypothesis testing earlier in this section, when we looked at null and alternative hypotheses. Hypothesis testing is a method for testing a hypothesis or claim about a parameter value, based on data from a sample, that helps in drawing conclusions about a population. The first (tentative) assumption is known as the null hypothesis. Then, an alternative hypothesis, which is the opposite of the null hypothesis, is defined.

To put these hypotheses into context, let’s consider an example. Say that an organization is performing an analysis of salaries of salespeople. It takes a sample of 100 salaries from a population of 10,000. The null hypothesis is that the mean salary of a salesperson is less than or equal to $85,000, and the alternative hypothesis is that the mean salary of a salesperson is more than $85,000.

The null hypothesis is usually represented as H₀, and the alternative hypothesis is usually represented as H_a. H₀ (the null hypothesis) is where things are happening as expected, and there is no difference from the expected outcome. H_a (the alternative hypothesis) is where things change from expected, and you have not just rejected H₀ but made a discovery.

Figure 10.6 gives an overview of the test statistic locations and their results.

In the context of our example of salesperson salaries, the sample mean (µ) is:

H₀: µ ≤ $85,000

H_a: µ > $85,000

There are various methods to perform hypothesis testing, such as by using Z-scores, t-tests, and p-values, as discussed earlier in this chapter.

Because hypothesis tests are based on sample information from a larger population, there is always a possibility of errors. To determine if the null hypothesis should be rejected, the hypothesis test should consider the level of significance for the test. The level of significance is represented as α and is commonly set as α = 0.05, α = 0.01, or α = 0.1. When working with hypotheses, errors are broadly classified as two types:

• Type I error: A type I error occurs when the null hypothesis is incorrectly rejected when, in fact, it is true. This is also known as a false positive.

• Type II error: A type II error occurs when the null hypothesis is not rejected (fail to reject) when, in fact, it is false. This is also referred to as a false negative.

Simple Linear Regression

Simple linear regression helps describe a relationship between two variables through a straight-line equation that closely models the relationship between these variables. This line, which is sometimes called the line of best fit, is plotted as a scatter graph between two continuous variables X and Y, where:

X is regarded as the explanatory, or independent, variable

Y is regarded as the response, or dependent, variable

For example, variable X could represent sales for a product Y (another variable), and as time passes, sales may go up or down, depending on how popular the product is (that is, the trend). Another example could be the speed (variable X) that the car would go and the miles (variable Y) per gallon (mpg) that the car owner expects. The mileage may vary depending on the speed of the vehicle and show a trend.

The formula to calculate simple linear regression is:

y = mx + b

where:

m is the slope

x is the mean

b is the intercept

Consider an example with the following sample values across the X and Y axes:

X = 100, 200, 300, 400, 500, 600

Y = 200, 300, 400, 500, 600, 700

In this case:

Mean X (MX) = (100+200+300+400+500+600) / 6 = 2100 / 6 = 350

Mean Y (MY) = (200+300+400+500+600+700)/6 = 2700 / 6 = 450

Sum of squares (SSq) = 175,000

Sum of products (Sp) = 175,000

The regression equation is:

y = bX + a

where:

b = Sp / SSq = 175,000 / 175,000 = 1

a = (MY) – b(MX) = 450 – (1 × 350) = 100

y = 1x + 100

The linear regression in this case can be represented as a scatter plot, with the values along the X and Y axes (see Figure 10.7).

Correlation

Correlation is a statistical measure that denotes the degree to which two values or variables are associated or related. It is used to describe a relationship without indicating cause and effect. Correlation coefficients are used to measure the (linear) relationship between two variables. When both values increase together, the linear correlation coefficient is positive. When one value increases and the other value decreases, the correlation coefficient is negative.

An example of correlation would be fuel prices and inflation in the cost of groceries. An increase in fuel prices impacts grocery prices as it costs more to transport food from farms to retail shops; therefore, as fuel prices rise, grocery prices also rise. This is a good example of positive correlation. On the other hand, as inflation rises, the general spending on items of want (not need) decreases. For example, with rising fuel prices, spending on cosmetics tends to decrease. This is a good example of negative correlation.

If X and Y are the two variables, the correlation coefficient can be represented by the following equation:

ρ = cov(X,Y) / σXσY

where:

ρ is the linear correlation coefficient

σ is the standard deviation

cov is covariance, a measure of how the two variables change together

The correlation coefficient is supposed to be linear (that is, follow a line). It can have several different values:

• ρ = 1 is a positive correlation coefficient

• ρ = 0 indicates null/no correlation found between the two values/variables

• ρ = –1 is a negative correlation coefficient

Figure 10.8 illustrates correlation relationships, including the (perfect) negative, null/no, and (perfect) positive correlations.