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# Quantitative Variables

Have you ever thought of finding the number of male and female students in your college?

Or have you ever thought about measuring the weight or height of your classmates, or recording the ages of your classmates to determine who is the youngest or oldest in your class?

All these are forms of data that can be counted and/or measured and represented in a numerical form. In statistics, these data are called quantitative variables.

In this article, we are going to study deeper into quantitative variables and how they compare to another type of variable, the qualitative variables.

## Quantitative variables meaning

Quantitative variables are variables whose values are counted.

Examples of quantitative variables are height, weight, number of goals scored in a football match, age, length, time, temperature, exam score, etc.

## Qualitative variables in statistics

Qualitative variables (also known as categorical variables) are variables that fit into categories and descriptions instead of numbers and measurements. Their values do not result from counting.

Examples of qualitative variables include hair color, eye color, religion, political affiliation, preferences, feelings, beliefs, etc.

## Types of quantitative variables

Quantitative variables are divided into two types: discrete quantitative variables and continuous quantitative variables. Details and differences between these two types of quantitative variables are explained hereafter.

## Discrete quantitative variable

Discrete quantitative variables are quantitative variables that take values that are countable and have a finite number of values. The values are often but not always integers.

The best way to tell whether a data set represents discrete quantitative variables is when the variables are countable and the number of possibilities is finite.

Continuous quantitative variables are quantitative variables whose values are not countable.

A discrete quantitative variable is a variable whose values are obtained by counting.

A continuous quantitative variable is a variable whose values are obtained by measuring.

When you count the number of goals scored in a sports game or the number of times a phone rings, this is a discrete quantitative variable.

When you measure the volume of water in a tank or the temperature of a patient, this is a continuous quantitative variable.

The table below contains examples of discrete quantitative and continuous quantitative variables,

 Discrete quantitative variables Continuous quantitative variables Number of children per household Weight Number of students in a college Speed of cars in a race Number of goals scored in a football match Height Number of correct questions answered in exams Temperature Number of people who took part in an election Time Number of students in a school Density

Distinguish the types of the following variables between discrete and continuous.

• Time taken for an athlete to complete a race,
• Depth of a river,
• Numbers of students present at school,
• Number of pets owned,

Solution

Continuous variables.

• The time taken for an athlete to complete a race, in order to see this, let us think of this situation as if we start a watch for an athlete to complete a 5000m race. From the start of the watch to the end of the race, the athlete might take 15 minutes:10 seconds:3milliseconds:5microseconds and so on depending on the precision of the stopwatch. This makes it a continuous variable.
• Depth of a river: a river may be 5m:40cm:4mm deep. Thus, the depth of a river is a continuous variable.

Discrete variables.

• Number of students present at school: this is discrete because it will always involve direct whole numbers in counting the number of students in school. We can have 1, 2, 3, 4, ...............200 students for instance present at school with a consistent interval of +1. We can never have 5.5 students or anything like that at any point. This makes it a discrete variable.
• The explanation above applies to the number of pets owned.

## Similarities between quantitative variables and qualitative variables

Primary data is the data collected by a researcher to address a problem at hand, which is classified into qualitative data and quantitative data.

Qualitative variables deal with descriptions that can be noticed but not calculated.

Quantitative variables focus on amounts/numbers that can be calculated.

✓ Both quantitative and qualitative data are used in research and analysis.

✓ Both are used in conjunction to ensure that the data gathered is free from errors.

✓Both can be obtained from the same data unit. Only their variables are different, i.e. numerical variables in case of quantitative data and categorical variables in case of qualitative data.

### Differences between quantitative and qualitative variables

 Quantitative variable Qualitative variable Can be counted and expressed in numbers and values. Cannot be counted but contains a classification of objects based on attributes, features, and characteristics. The research methodology is conclusive in nature and aims at testing a specific hypothesis to determine the relationships. The research methodology is exploratory, that is it provides insights and understanding. Has a focused approach and is objective. The research approach is subjective. Uses statistical analysis methods of analysis. The analysis is non-statistical. Ascertains the level of occurrence. Determines the depth of understanding Sample size is large and drawn from the representative sample. The sample size is usually small and is drawn from non-representative samples. Methods of data collection include experiments, surveys, and measurements. Methods of data collection include interviews, focus groups, observation, and archival materials like newspapers. Examples include height, weight, age, exam scores, etc. Examples include opinions, beliefs, eye color, description, etc.

Determine if the following variables are quantitative or qualitative variables,

• hair color
• time
• gender
• distance in kilometers
• temperature
• music genre

Solution

Qualitative variables.

• Hair color: hair colors can be grouped into various categories; whether you have blonde hair, brunette, red, or black. In a family of 5 people, 2 may have blonde hair, 2 may be brunette, 1 red, and 0 black and we can classify the people according to their hair colors. Therefore it is a categorical variable.
• Gender: this is a categorical variable because obviously, each person falls under a particular gender based on certain characteristics. A person may be a male, female, or fall under any other gender category. If there are 20 workers in a company and we want to group them according to gender, we may have 15 females and 5 males. This makes gender a qualitative variable.
• Music genre: there are different genres to classify music. Either Jazz, Rock, Hip hop, Reggae, etc.

Quantitative variables.

These are the variables that can be counted or measured.

• time in minutes: it might take a student 10 hours to finish studying this topic. Here, we are interested in the numerical value of how long it can take to finish studying a topic. This makes the time a quantitative variable.
• Temperature in degrees Celsius: the temperature of a room in degrees Celsius is a quantitative variable as it is measured and recorded in numerical as say 25, 26, or 30 degrees Celsius.
• Distance in kilometers: this is also quantitative as it requires a certain numerical value in the unit given (kilometers).

Note that the distance as a quantitative variable is given in kilometers or measurable units otherwise distance may be described as short, long, or very long which then will make the variable qualitative/categorical.

## Quantitative variables representation

Quantitative variables can generally be represented through graphs. There are many types of graphs that can be used to present distributions of quantitative variables.

✓ Stem and leaf displays/plot. A graphical type of display used to visualize quantitative data. Stem and leaf plots organize quantitative data and make it easier to determine the frequency of different types of values.

✓ Histograms. A type of graph that summarizes quantitative data that are continuous, meaning they a quantitative dataset that is measured on an interval. Histograms represent the distinctive characteristics of the data in a user-friendly and understandable manner.

✓ Frequency polygons. A line graph used for a visual representation of quantitative variables. Frequency polygons indicate shapes of distributions and are useful for comparing sets of data. In this type of data visualization, the data are plotted on a graph and a line is drawn connecting points to each other to understand the shape of the variables.

✓ Box plots. A graphical representation method for quantitative data that indicate the spread, skewness, and locality of the data through quartiles. Box plots are also known as whisker plots, and they show the distribution of numerical data through percentiles and quartiles.

✓ Bar charts. A graph in the form of rectangles of equal widths with their heights/lengths representing values of quantitative data. A bar graph/chart makes quantitative data easier to read as they convey information about the data in an understandable and comparable manner. The horizontal axis of a bar graph is called the y-axis while the vertical axis is the x-axis. Bar graphs make a comparison between data easier and more understandable.

✓ Line graphs. This is a line or curve that connects a series of quantitative data points called ‘markers’ on a graph. Similar to box plots and frequency polygons, line graphs indicate a continuous change in quantitative data and track changes over short and long periods of time.

✓ Scatter plots. Scatter plots use cartesian coordinates to show values for two variables for a set of data. Scatter plots basically show whether there is a correlation or relationship between the sets of data.

Note that some graph types such as stem and leaf displays are suitable for small to moderate amounts of data, while others such as histograms and bar graphs are suitable for large amounts of data. Graph types such as box plots are good when showing differences between distributions. Scatter plots are used to show the relationship or correlation between two variables.

## Quantitative Variables - Key takeaways

• Quantitative variables are variables whose values result from counting or measuring something.
• Quantitative variables are divided into two types: discrete and continuous variables.
• Discrete variables take values that are countable and have a finite number of values.
• Continuous variables are variables whose values are not countable and have an infinite number of possibilities.
• Examples of methods for presenting quantitative variables include Stem and leaf plots, histograms, frequency polygons, box plots, bar charts, line graphs, and scatter plots.

## Frequently Asked Questions about Quantitative Variables

Examples of quantitative variables are height, weight, number of goals scored in a football match, age, length, time, temperature, exam score, etc.

The three types of quantitative variables are discrete, continuous, and mixed quantitative variables

Quantitative variables are variables whose values are counted.

Quantitative variables are variables whose values are counted.

Quantitative variables can be counted and expressed in numbers and values while qualitative /categorical variables cannot be counted but contain a classification of objects based on attributes, features, and characteristics.

## Final Quantitative Variables Quiz

Question

What are quantitative variables?

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Answer

Quantitative Variables are variables whose values result from counting or measuring something

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Question

Qualitative Variables are variables that fit into categories and descriptions instead of measurements or numbers. True/False

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Answer

True

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Quantitative variables are divided into two types, these are:

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Discrete variables and continuous variables

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They are quantitative variables whose values are not countable and have an infinite number of possibilities.

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Continuous variables

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A continuous variable is a variable whose value is obtained by counting. True/False

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Answer

False

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Question

Quantitative variables can be represented in several graph forms including………

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Stem and leaf displays/plots, histograms, frequency polygons, box plots, bar charts, line graphs, and scatter plots

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Question

The research approach for qualitative data is subjective and holistic. True/False

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True

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Question

Compared to qualitative research methodology which is exploratory, quantitative research methodology is……………

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conclusive in nature and aims at testing a specific hypothesis to determine the relationships

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A similarity between qualitative and quantitative data is…………

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Both quantitative and qualitative data could be used in research and analysis

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The three data analysis methods for quantitative data are …………

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Cross-tabulation, Trend analysis, and Conjoint analysis

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The analysis method that compares data collected over a period of time with the current to see how things have changed over that period is……..

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Trend analysis

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Examples of quantitative variables include........

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Height, weight, number of goals scored in a football match, age, length, time, temperature, exam score, etc

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Quantitative variables are divided into _________

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Discrete (categorical) and continuous variables

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A suitable graph for presenting large amounts of distributions of quantitative data is the _______________

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Histogram

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Small to moderate amounts of quantitative data can be best represented using_______

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Stem and leaf displays

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When showing differences between distributions, the best diagram to use is the____

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Box plot

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A _________is the  suitable graph to be used to show the relationship (correlation) between two variables.

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Scatter plot

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Both discrete and continuous variables are        ___________

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Answer

Quantitative variables

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Both quantitative and qualitative data can be classified as ____________

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Primary data

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Two main types of variables are ____________

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Quantitative variables and Qualitative variables

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Quantitative variables can be categorized as

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Discrete and continuous

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Focus Group,Observation, Interviews,Archival Materials are ________

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Methods of collecting qualitative data

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Experiments,Surveys and Observations   Methods used for collecting data for_______

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Quantitative research

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A method of quantitative data analysis that analyzes the relationship between multiple variables is known as____

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Cross-tabulation

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A method of quantitative data analysis that

compares data collected over a period of time with the current to see how things have changed over that period is known as ______________

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Trend analysis

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This type of quantitative analysis method assigns values to different characteristics and ask respondents to evaluate them.

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Conjoint analysis

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Age,weight,height temperature etc. are examples of ___________

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Continuous variables

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What does a box plot show us?

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Answer

The spread of our data that can be interpreted with our five number summary.

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What are the five numbers of our five number summary?

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• Minimum
• Maximum
• Median (Q2)
• Lower Quartile (Q1)
• Upper Quartile (Q3)

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True or False.

The order of your numbers does not matter?

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Answer

False. Numbers must be ordered from least to greatest.

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Question

In the following data set which numbers are the minimum and maximum:

7  12  3  15  17  19  21

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Answer

7 and 21

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How do you find the median (Q2) of your data?

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Answer

Add the numbers and divide by 2

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True or False.

When finding the lower quartile (Q1) and upper quartile (Q3) you do not include the median (Q2) value

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Answer

True. The median (Q2) is not included in this step.

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Question

Determine Q1 for the following data set:

3  5  8  10  12  12  15

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Answer

10

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Determine the Q3 for the following data set

27  30  35  42  45

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Answer

42.5

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IQR stands for what?

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Answer

Interquartile Range

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How do you calculate range?

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Answer

maximum - minimum

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If I have the following what have I just found?

Q3 = 20

Q1 = 7.5

Q3 - Q1 = 20 - 7.5 = 12.5

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Answer

Range

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The Interquartile Range tells us what?

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Where the middle 50% of our data is

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True or False.

The following data contains an outlier:

14  17  22  22  22  25  36

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Answer

False. The upper range is 37 and the lower range is 5. All values fall within the normal range.

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Variation means?

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Answer

How varied, or spread out, our data is.

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What is standard deviation?

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Answer

Standard deviation is a measure of the spread of a data-set.

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What is a sample data set?

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Answer

A sample data set is a data set that includes a representative fraction of a specified group.

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What is a population data set?

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A population data set is a data set that includes all members of a specified group.

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What is the mean of a data set?

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The mean of a data set is it's average value.

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What is the formula for the mean of a data set?

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Answer

$\mu = \frac{\displaystyle \sum_{i=1}^N x_{i}}{N}$

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What is the formula for the standard deviation of a population data set?

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$\sigma = \sqrt{\frac{\displaystyle \sum_{i=1}^N (x-\mu)^2}{N}}$

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What is the formula for the standard deviation of a sample data set?

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$\sigma = \sqrt{\frac{\displaystyle \sum_{i=1}^N (x-\bar{x})^2}{N-1}}$

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What is the other name for the empirical rule?

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$$68$$-$$95$$-$$99.7$$ rule

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What does the empirical rule state?

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The empirical rule states that for most normally distributed data sets, $$68\%$$ of data points are within one standard deviation of the mean, $$95\%$$ of data points are within two standard deviations of the mean, and $$99.7 \%$$ of data points are within three standard deviations of the mean.

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