## Objectives

By the end of this lesson, you will be able to...

- describe what "statistics" means in the context of this course.
- explain the process of statistics
- distinguish between qualitative and quantitative variables
- distinguish between discrete and continuous variables
- determine the level of measurement of a variable

For a quick overview of this section, watch this short video summary:

The first thing we want to look at is exactly what "statistics" is. One reasonable definition might be:

**Statistics** is the science of collecting, organizing, summarizing, and analyzing information to draw conclusions or answer questions.

So what does that mean? Well, the reason we use data is that often the anecdotal information we have which might *appear* to be true actually is not.

Case in point: Recently the math department at ECC decided to do some investigation into the success of students in the College Algebra course. Many instructors had poor experiences with students who placed into the course with an ACT score of 23. Those faculty members felt that the cut-off score for placement into College Algebra should be increased to at least 24.

An interesting thing happened when data was collected from a year's worth of students - the students with a 23 on their ACT did just as well as students who placed their via ECC's own placement exam or those who took Intermediate Algebra first. Oops! This was a reminder to even those of us in the math department that there's a reason why the study of statistics was developed - we often have a skewed sense of reality when we only trust our experiences.

At this point, I'd strongly recommend beginning a list of terms with definitions. You might start to get overwhelmed with all the terminology, so a list of terms to refer to would be very helpful.

## The Process of Statistics

So what exactly is the study of statistics? Well, it's really a process.

First, we must identify exactly what it is we're hoping to study. We must also determine what our population is. | Next, we select a representative sample using appropriate sampling techniques. | Once we have our data collected, we have to summarize it. We'll do this both numerically and visually with charts. | Finally, we need to analyze it and come to a conclusion. |

The gaps in the middle - Chapters 4-8 - are a mix of sections. Chapter 4 really can stand on its own. It's all about analyzing the relationship between two variables. Chapters 5-8 involve probability and are intended as preparation for the meat of the course in Chapters 9-12.

## Identifying the Question

A couple of key comments about identifying the question are needed here. The first thing we really need to consider is what our population is. The **population** is the group we're studying.

For example, if I'm interested in the studying habits of ECC students, then my population is *all ECC students*. Since asking every ECC student isn't possible, I would then take a **sample**, which is a subset of the population. The characteristics of the sample are key. If we select too few or the individuals selected don't represent the population, any conclusions we draw will be meaningless.

A **statistic** is a numerical summary of a sample. By contrast, a numerical summary of a *population* is called a **parameter**.

For example, if we know from ECC data that the average age of all ECC students is 29 (ECC College Facts), that value is a *parameter*. On the other hand, if we take a sample of 100 students and find that 63% support a new initiative at the college, that is a *statistic* - since it is only a measure of the sample of 100 students, not the entire student population.

When we simply describe or summarize data, we're using **descriptive statistics**. When we draw conclusions or extend our results to the population, we're using **inferential statistics**.

For example, the statistics of 63% from above would be a *descriptive statistic*, since it is simply a summary of our sample. If we, in turn, make a broad generalization and claim that 63% of *all* ECC students support the initiative, then that is *inferential statistics*.

## Qualitative or Quantitative

In general, we classify data into two groups: *qualitative* or *quantitative*.

**Qualitative (or categorical) variables** allow for classification of individuals based on some attribute or characteristic.

**Quantitative variables** provide numerical measures of individuals. Arithmetic operations such as addition and subtraction can be performed on the values of a quantitative variable and will provide meaningful results.

Basically, if a variable describes a *quality* of an individual - i.e. hair color, political party, etc - then it is *qualitative*. If a variable is numerical **and those numbers have meaning**, then it is *quantitative*. (Not all numbers have meaning numerically - think of an individuals Social Security number.)

Example 1

So, which are they? Here are some examples of data that might be collected. Take a minute and make a note of whether each is qualitative or quantitative. When you're ready, check your answer below.

gender, IQ, ACT score, eye color, area code

[ reveal answer ]

**Gender - qualitative. **No numbers here, so it's an easy choice.

**IQ - quantitative.** IQs are numbers and can be averaged - with meaning.

**ACT score - quantitative. **Similar reasoning to IQs.

**Eye color - qualitative.** Same reasoning as gender.

**Area code - qualitative. **While they are numbers, you can't have an *average* area code.

## Discrete or Continuous

Quantitative variables can be further split into two groups.

A **discrete variable** is a quantitative variable that has either a finite number of possible values or a countable number of values. (*Countable* means that the values result from counting - 0, 1, 2, 3, ...)

A **continuous variable** is a quantitative variable that has an infinite number of possible values that are not countable.

Most variables are pretty clear, but some can be a bit tricky. An example of a tricky one is time. Say, for example, we're looking at how long we've been waiting for a bus. We *count* the minutes and seconds, but really those time units are only rounded. There are actually milliseconds, nanoseconds, etc - an infinite number of possibilities in the middle. So actually, any variable that is time is continuous.

Here's a graphical representation of the different ways to classify variables:

Example 2

Time for some examples. Take a minute and make a note of whether each quantitative variable is discrete or continuous. When you're ready, check your answer below.

IQ, ACT score, height, distance commuting, shoe size

[ reveal answer ]

**IQ - discrete.** IQ scores are always integers - 100, 110, 180, etc.

**ACT - discrete. **Same reasoning as IQ.

**Height - continuous.** Even though your height might be 5'8", it's really 5.68241231... feet. It's impossible to measure a length exactly!

**Distance commuting - continuous.** Similar reasoning to height.

**Shoe size - discrete. **This is a tough one. An argument could be made for either choice, but shoe sizes only come in whole numbers or possibly 1/2 sizes - there is no 8.24 shoe size.

## Level of Measurement

Instead of categorizing variables into qualitative or quantitative, we can assign them various "levels" based on their characteristics.

**Nominal:** A **nominal** variable is one that simply categorizes or names the variables (i.e. "hair color"). This is the most general level of measurement.

**Ordinal:** An **ordinal** variable categorizes, but also has a specific order, like "course grade". This is a little more specific than a nominal variable.

**Interval:** An **interval** variable is quantitative (so the values have order as numbers), the differences between values have meaning, but a value of 0 doesn't mean the object has no value. The easiest example is "temperature" - clearly 60°F is more than 30°F, but 0°F doesn't mean it has no temperature and couldn't get colder.

**Ratio:** The final and most precise level of measurement is **ratio**. A ratio variable has all the properties of an interval variable, but the ratio of two values has meaning, and a value of zero means the absence of that quantity. A simple example might be "points earned on an exam". A score of 80 is twice the value of a score of 40, and a score of 0 clearly means the student earned no points on the exam. (Note that this is different than a grade of A vs. F, which would just be ordinal.)

Example 3

Time for one last set of examples. Categorize each variable based on its level of measurement as nominal, ordinal, interval, or ratio.

gender, IQ, distance commuting, pain rating 1-10

[ reveal answer ]

**Gender - nominal**. This is just categorizing each individual.

**IQ - interval**. Larger values do imply a higher IQ, but a 0 IQ doesn't have any meaning - that individual is dead! Also, an individual with an IQ of 120 isn't really "twice as smart" as an individual with an IQ of 60.

**Distance commuting - ratio**. Unlike IQ, it's possible to have a distance commuting of 0 (working at home!), and a ratio of two distances does have meaning.

**Pain rating - ordinal**. While it may seem that this is an interval or even a ratio variable, that's not the case. Clearly 7 is more than 5 and 5 is more than 3, but does that difference of 2 mean the same in both cases? What about the difference between a 10 and an 8? Because those differences aren't consisten, this is not an interval variable.

## FAQs

### What is 1 Introduction to statistics? ›

Statistics is **the study of the collection, analysis, interpretation, presentation, and organization of data**. In other words, it is a mathematical discipline to collect, summarize data. Also, we can say that statistics is a branch of applied mathematics.

**What are the 3 types of statistics? ›**

They are: **(i) Mean, (ii) Median, and (iii) Mode**. Statistics is the study of Data Collection, Analysis, Interpretation, Presentation, and organizing in a specific way.

**What are the 4 basic elements of statistics? ›**

**Sample size, variables required, numerical summary tools, and conclusions** are the four elements of a descriptive statistics problem.

**Is Intro to Stats hard in college? ›**

Statistics is a very important subject that every student in their undergrad should take regardless of their major. **It may be difficult at first, but it is just like learning a new language; once the basics are understood and practiced, it becomes much easier and almost second nature over time.**

**How hard is statistics in college? ›**

It's a challenging course with a lot of concepts mixed with algebra and reading comprehension. And unfortunately for some students, they will struggle to keep up with the rigor of a college-level stats class.

**What kind of math is in statistics? ›**

Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on **differential and integral calculus, linear algebra, and probability theory**.

**What are the 2 main types of statistics? ›**

**Statistics have majorly categorised into two types:**

- Descriptive statistics.
- Inferential statistics.

**What are the 2 main methods in statistics? ›**

Two main statistical methods are used in data analysis: **descriptive statistics, which summarizes data using indexes such as mean and median and another is inferential statistics**, which draw conclusions from data using statistical tests such as student's t-test.

**What are the 5 basic words of statistics? ›**

The five words **population, sample, parameter, statistic (singular), and variable** form the basic vocabulary of statistics.

**What are the 5 main statistics? ›**

A summary consists of five values: **the most extreme values in the data set (the maximum and minimum values), the lower and upper quartiles, and the median**. These values are presented together and ordered from lowest to highest: minimum value, lower quartile (Q_{1}), median value (Q_{2}), upper quartile (Q_{3}), maximum value.

### How can I pass statistics? ›

**The Five Steps to Passing Statistics**

- Refreshing your knowledge of foundational concepts.
- Mastering statistics fundamentals.
- Using your time wisely.
- Getting help early if you need it.
- Not stressing about the course.

**Is intro to statistics harder than college algebra? ›**

Is statistics easier than college algebra? Algebra concepts are much easier to grasp, Stats concepts are harder to grasp but the work itself at an INTRO level stat class will be easier as most of it is just memorizing a bunch of formulas and plugging them in. Anything above intro stats would require knowing calculus.

**Are statistics easy? ›**

Stats is hard and involves a lot of math. If you're in a different field than math, the stats courses are usually rote memorization, rather than understanding of why we end up with those equations.

**Is stats harder than math? ›**

At an advanced level, statistics is considered harder than calculus, but beginner-level statistics is much easier than beginner calculus. Frankly, it mostly depends upon the student's interest as some students find it hard to comprehend statistics while others find it hard to understand calculus.

**Is stats harder than calculus? ›**

Probability and statistics requires a slightly different way to look at things. For most students it is more difficult than calculus.

**Do colleges prefer stats or calculus? ›**

But for many other students, calculus isn't the math course that will most help them—the right course often is statistics. But **most admissions counselors have favored calculus (in many cases informally)**, the report says, and that hurts students.

**How many people fail statistics in college? ›**

College dropout rates indicate that **up to 32.9%** of undergraduates do not complete their degree program. First-time undergraduate freshmen have a 12-month dropout rate of 24.1%. Among first-time bachelor's degree seekers, 25.7% ultimately drop out; among all undergraduate students, up to ^{40}^{%} drop out.

**Does statistics require a lot of math? ›**

**While the leading Mathematicians think of Statistics as no more than a part of Applied Mathematics, many think otherwise**. Both subjects are complementary and use similar methodologies. To understand statistical techniques better, it is helpful to have a strong grasp of math.

**What is the hardest topic in statistics? ›**

The most difficult topic in statistical inference is the '**Test of hypothesis**. ' The point where one has to actually figure out the null and alternative hypotheses is one of the crucial points.

**Is statistics same as calculus? ›**

**Calculus and statistics both center on models of relationships**: constructing them, analyzing them, evaluating them. In calculus, the choice to add a term to a model reflects some knowledge or hypothesis about mechanism. In statistics, choices are based on evidence provided by data.

### Is statistics required in college? ›

**Statistics is specifically required in some majors, while it is a quantitative methods requirement in others**.

**Why is college math so hard? ›**

“The sequential nature of math coupled with its own vocabulary, need for persistent studying, and the speed at which math is taught in higher education, with approximately 15 weeks in a semester, creates major problems for college students.” All of this mathematical jargon can be tough to retain, so it's important to ...

**What are examples of statistics? ›**

A statistic is a number that represents a property of the sample. For example, if we consider one math class to be a sample of the population of all math classes, then **the average number of points earned by students in that one math class at the end of the term** is an example of a statistic.

**What is the purpose of statistics? ›**

The main purpose of using statistics is to plan the collected data in terms of experimental designs and statistical surveys. Statistics is considered a mathematical science that works with numerical data. In short, statistics is a crucial process which helps to make the decision based on the data.

**How is statistics used in everyday life? ›**

**Individuals use statistics to make decisions in financial planning and budgeting**, while organizations are guided by statistics in financial policy decisions. Banks use statistics to lower risk in lending operations, analyze activity in the financial market, and predict the impact of economic crises.

**Who is father of statistics? ›**

**Prof.** **Prasanta Chandra Mahalanobis** is also known as the father of Indian Statistics.

**What is the simple definition of statistics? ›**

statistics, **the science of collecting, analyzing, presenting, and interpreting data**.

**What is statistics in your own words? ›**

Statistics is **the science concerned with developing and studying methods for collecting, analyzing, interpreting and presenting empirical data**.

**What are the 3 components of statistics? ›**

There are three real branches of statistics: **data collection, descriptive statistics and inferential statistics**.

**What is the most important in statistics? ›**

The three essential elements of statistics are measurement, comparison, and variation. Randomness is one way to supply variation, and it's one way to model variation, but it's not necessary.

### What are the formulas of statistics? ›

Mean | x ¯ = ∑ x n | x = Observations given n = Total number of observations |
---|---|---|

Variance | σ 2 = ∑ ( x − x ¯ ) 2 n | x = Observations given x ¯ = Mean n = Total number of observations |

Standard Deviation | S = σ = ∑ ( x − x ¯ ) 2 n | x = Observations given x ¯ = Mean n = Total number of observations |

**What are the key statistics? ›**

Key Statistics are **important points of financial data that allow you to quickly view and ascertain transaction history, as well as provide insight into donation trends**. These statistics can be found both on the Dashboard as entire database summaries and on an individual basis at the top of each entity record.

**What all topics are in statistics? ›**

**Mathematical Statistics Topics**

- Combinatorics and basic set theory notation.
- Probability definitions and properties.
- Common discrete and continuous distributions.
- Bivariate distributions.
- Conditional probability.
- Random variables, expectation, variance.
- Univariate and bivariate transformations.

**How do I study for statistics? ›**

**Study Tips for the Student of Basic Statistics**

- Use distributive practice rather than massed practice. ...
- Study in triads or quads of students at least once every week. ...
- Don't try to memorize formulas (A good instructor will never ask you to do this). ...
- Work as many and varied problems and exercises as you possibly can.

**What are the 6 methods of collecting data in statistics? ›**

The most commonly used methods are: **published literature sources, surveys (email and mail), interviews (telephone, face-to-face or focus group), observations, documents and records, and experiments**.

**What is the most important step in statistics? ›**

**Plan (Ask a question**): formulate a statistical question that can be answered with data. A good deal of time should be given to this step as it is the most important step in the process.

**Why is statistics so difficult? ›**

Why is statistics so hard? **There are a lot of technical terms in statistics that may become overwhelming at times**. It involves many mathematical concepts, so students who are not very good at maths may struggle. The formulas are also arithmetically complex, making them difficult to apply without errors.

**How do Beginners study statistics? ›**

**Here are some tips to make your learning more efficient and fruitful.**

- 1) Learn the core mathematics first, then the statistics. ...
- 2) Learn about what statistics can do, not about what it can say. ...
- 3) Probability theory and statistics go hand in hand. ...
- 4) Regression analysis is very useful, but also often misused.

**What is a passing grade in stats? ›**

This is an above-average score, between 80% and 89% C - this is a grade that rests right in the middle. C is anywhere between 70% and 79% D - this is still a passing grade, and it's **between 59% and 69%**

**Is statistics easier than AP calculus? ›**

**Many students find AP Statistics next to calculus in terms of difficulty**, with lower pass rates and fewer perfect scores than those of other AP courses. Even so, passing the AP Statistics exam can lead to advanced placement and even college credit for science, math, engineering, and criminal justice majors.

### Is statistics harder than AP calculus? ›

**The content covered in AP Statistics is generally considered easier and more manageable than that of the two AP Calculus exams**. Many students have learned some statistical concepts in previous math classes, and they often find the concepts easier to understand than other math subjects such as calculus or geometry.

**Do I need to be good at algebra for statistics? ›**

**Basic statistics requires at least a good foundation in algebra**. A first semester of calculus would be helpful too. That being said, most if not all of the statistical techniques you will be learning will usually not be done by hand.

**Is statistics 1 harder than calculus? ›**

**At an advanced level, statistics is considered harder than calculus**, but beginner-level statistics is much easier than beginner calculus. Frankly, it mostly depends upon the student's interest as some students find it hard to comprehend statistics while others find it hard to understand calculus.

**How do you pass an intro to statistics? ›**

**The Five Steps to Passing Statistics**

- Refreshing your knowledge of foundational concepts.
- Mastering statistics fundamentals.
- Using your time wisely.
- Getting help early if you need it.
- Not stressing about the course.

**How do you study statistics intro? ›**

**Study Tips for the Student of Basic Statistics**

- Use distributive practice rather than massed practice. ...
- Study in triads or quads of students at least once every week. ...
- Don't try to memorize formulas (A good instructor will never ask you to do this). ...
- Work as many and varied problems and exercises as you possibly can.

**What is statistics in simple words? ›**

: **a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data**. : a collection of quantitative data.