What Is Discrete Data?


What Is Discrete Data

What is the meaning of discrete data?

Discrete data is information that can only take certain values. These values don’t have to be whole numbers (a child might have a shoe size of 3.5 or a company may make a profit of £3456.25 for example) but they are fixed values – a child cannot have a shoe size of 3.72! The number of each type of treatment a salon needs to schedule for the week, the number of children attending a nursery each day or the profit a business makes each month are all examples of discrete data.

This type of data is often represented using tally charts, bar charts or pie charts. Continuous data is data that can take any value. Height, weight, temperature and length are all examples of continuous data. Some continuous data will change over time; the weight of a baby in its first year or the temperature in a room throughout the day.

This data is best shown on a line graph as this type of graph can show how the data changes over a given period of time. Other continuous data, such as the heights of a group of children on one particular day, is often grouped into categories to make it easier to interpret.

What is continuous and discrete data?

Difference between discrete data vs continuous data – Both discrete data and continuous data forms are crucial for statistical analysis plan, However, before making any inferences or assumptions about the specific data type, it is important to comprehend some significant differences between the two. So let’s find out the differences between discrete data vs continuous data:

Discrete Data Continuous Data
Takes particular countable values. Takes any measured value within a given range.
Discrete data is information that has noticeable gaps between values. Continuous data is information that occurs in a continuous series.
Discrete data is made up of discrete or distinct values. Directly in opposition, continuous data includes any value that falls inside a range.
Discrete data can be counted. Continuous data is quantifiable.
Bar graphs are a visual representation of discrete data. Continuous data are graphically represented using a histogram.
Ungrouped frequency distribution refers to the tabulation of discrete data against a single value. The tabulation of continuous data performed against a set of values is called grouped frequency distribution.
For discrete data, a classification like 10-19, 20-29, etc., are non-overlapping or mutually inclusive. For continuous data, classifications such as 10-20, 20-30, etc., overlap or are mutually exclusive.
The discrete function graph exhibits a distinct point that is nonetheless disconnected. A broken line connects the points on a continuous function graph.
Examples of frequent discrete data include the number of students, children, shoe size, and so forth. Some common continuous data types are height, weight, time, temperature, age, etc.

What is discrete vs non discrete?

Details Category: Part Record It is important to know the difference between Discrete and Non-Discrete Parts when you configure the system. On the advanced tab of the Part Definition Form you can define if Part Records will be Discrete or Non-Discrete.

Discrete Parts Discrete Parts are Part Records that can only have a quantity of 1. It is best to use the ‘Discrete’ option on the Part Definition if you want to have individual traceability for single tangible parts. You can make parts traceable by the NV Unique ID (Serial Numbers) Some examples:

Any part with an individual traceability requirement (anything with a serial number). Any other part can be defined as Non-Discrete.

Non-Discrete Parts Non-Discrete Parts are Part Records that can have any quantity larger than zero. It is best to use the ‘Non-Discrete’ option on the Part Definition if you have parts that do not need individual traceablity or if you want trace batches of parts. Some examples:

A batch of glue A bag of screws where you want to track just the part count, batch number and/or supplier Any other parts where traceability is not a requirement

Is Age discrete or continuous?

The exact age is a continuous variable, but age is often rounded down to the closest integer.

How do you know if data is discrete?

What is Discrete Data? – Discrete data also referred to as discrete values, is data that only takes certain values. Commonly in the form of whole numbers or integers, this is data that can be counted and has a finite number of values. These values must be able to fall within certain classifications and are unable to be broken down into smaller parts. Some examples of discrete data would include:

The number of employees in your department The number of new customers you signed on last quarter The number of products currently held in inventory

All of these examples detail a distinct and separate value that can be counted and assigned a fixed numerical value.

Is population discrete or continuous?

So to answer your question, it considered continuous in the sense of the first definition. Population densities are ratios and therefore, have values that vary continuously, unlike population counts which have values that vary in discrete increments.

Is gender discrete or continuous?

Chapter 7 Variables Associations We can “get a sense” if a discrete and a continuous variable seem associated visually through a chart called a boxplot (discussed further below) and numerically through examining the difference of means (or medians, if one so prefers).

What type of an association do we get when we consider a discrete and a continuous variable? The easiest way to represent this type of association is when we consider a binary (two-category) discrete variable and check if a continuous variable’s statistics (like the mean, or the median) vary between the discrete variable’s categories.

This sounds far more complicated than it is. A couple of examples will show you that you have probably considered questions about “comparisons of means” even in your everyday life. The first one will explain it conceptually, the second with actual data.

Research has shown that, despite similar lower body strength, women have less upper body strength than men, on average. One such study examined differences in upper body strength in a sample of Caucasian and East-Asian college students engaged in weight-lifting classes in American colleges (Chen, Liu and Yu, 2012),

While the study examined numerous aspects of the difference in strength, I will take only one of the researchers’ findings to illustrate my point: triceps strength in arm extension. The reported means were 46.2 pounds for women versus 87.4 pounds for men in the Caucasian sample, and 39.6 pounds for women versus 82.1 pounds for men in the East-Asian sample (Chen, Liu and Yu, 2012, p.156).

  1. Consider what we are discussing here: We have two variables of interest, gender and upper-body strength,
  2. Gender is a nominal discrete (and, in this study, binary) variable while upper-body strength (through various measurements in pounds) is a ratio continuous variable.
  3. The hypothesized association between the two posits that some categories of the discrete variable (e.g., men) tend to go with specific values of the continuous variable (e.g., higher values on upper body-strength).

That is, if both men and women had the same means for, in this case, triceps strength in arm extension, gender and upper-body strength would be unrelated, as one’s sex wouldn’t be predictive of one’s upper-body strength at all. In effect, we are comparing the mean values (of a continuous variable) across groups (i.e., the categories of a discrete variable). (Caucasian sub-sample) (East-Asian sub-sample) Thus, what we observe in this sample is a 41.2 pounds difference in the upper-body strength (as measured by triceps strength in arm extension) between Caucasian men and women and a difference in upper-body strength of 42.5 pounds between East-Asian men and women.

Again, note that the fact that we see these differences in the sample does not mean they exist in the population — they may, or they might not, We wouldn’t know this unless we test if the differences are generalizable to the population, We will get to testing later, for now we are only interested in the differences descriptively, i.e., that they exist in the sample,

Example 7.1 above shows that every time we compare averages of two (or indeed, more than two) groups and calculate the differences in the means, we are effectively describing associations between variables. I could have easily presented other examples like gender or race/ethnic differences in annual income, years of education, occupational prestige, test scores, etc., etc.

The reason I chose an example about a sex-based rather than gender-based difference (that is, a kinesiological rather than a sociological study) was so that I can warn you in passing about a common mistake, called the ecological fallacy, Consider the findings from the study in Example 7.1 above: men’s average upper-body strength is higher than women’s.

Assuming we can generalize the findings to the general population, the evidence suggests than when it comes to upper-body strength men are stronger than women on average, Many people take this to mean that a randomly selected man would be always stronger than a randomly selected woman,

  1. Which does not follow at all from the difference in mean strength.
  2. Statistically speaking, it is a matter of the dispersion around the means of the two groups, and of how big the difference in means is.
  3. It is quite possible for a lot of women to have more upper-body strength than the men’s average, as well as that a lot of men to have less upper-body strength than the women’s average.

Ultimately, the takeaway from this caveat is to not over-interpret differences in averages to mean more than what they actually are: differences in averaged values, not of the specific values of individuals belonging to the different groups that are compared.

You can find an excellent account of how common this ecological-fallacy mistake is here: https://www.americanscientist.org/article/what-everyone-should-know-about-statistical-correlation,) With that warning out of the way, let’s take another (this time, sociologically motivated) example for examining differences of means, along with a proper visual description — boxplots.

Statistics Canada’s National Household Survey 2011 (NHS 2011) was designed to replace the until-that-time mandatory long form of the Census, For this example, I use a random sample of about 3 percent of the NHS 2011 individual data (aka a Public Use Microdata File, or PUMF), resulting in N =22,123.

  1. I am interested in whether men and women’s income for the year preceding the survey differed, i.e., whether the variables gender (called sex in the dataset) and total income (i.e., income from all possible sources) appear associated.
  2. With the help of SPSS, I plot the data.
  3. The resulting boxplots graph is given in Figure 7.1 below.
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Figure 7.1 Gender Differences in Total Income, NHS 2011 What Is Discrete Data Boxplots are charts visually incorporating a lot of statistical information in one neat little package; I encourage you to make use of them when exploring your data as they can be quite useful. What do we see in Figure 7.1 in our case? Obviously, we have two groups to compare (as per the two categories in the nominal variable gender), women and men, and therefore the graph presents two boxplots.

(Had we multiple categories in our discrete variable, we’d have had multiple boxplots.) How to read a boxplot. Each boxplot consists of the eponymous “box” and two so-called “whiskers” protruding from it. The “box” (in green above) represents the middle 50 percent of the data (i.e., the two middle quartiles, or the IQR); the lower “whisker” represents the first/bottom quartile of the data, and the upper “whisker” represents the last/top quartile of the data.

The dark line bisecting the box indicates the median. The two ends of the “whiskers” are the lowest and the highest values. Note, however, that the quartiles (as represented by the “whiskers”) exclude outliers as to not visually distort the “regular” spread of the data.

As such, the chart plots run-of-the-mill outlier cases as small circles (above they are in red) outside of the “whiskers”; extreme outliers are indicated by stars (in black above), Now that you know how to read them, compare the two boxplots above. First, we see that the median for men is higher than the median for women (again, these are the dark lines within the boxes) ; as well, total income appears to be more spread out for men than for women (the “whiskers” in the men’s boxplot reach further, indicating larger range and IQR.

Further, while both men and women appear to have outliers, the men’s group seems to include more extreme outliers and at higher values than those observed in the women’s group, All this points to the conclusion that men in the sample had higher (median, and quite likely average) total income for 2010 than women did, despite that the individuals with the lowest incomes also appear to be men.

  • As useful the general information we gleaned from the boxplots, we should look at the precise numbers too.
  • SPSS calculates the mean total income as $32,465 for women and $48,866 for men — that is, there is $16,401 difference in mean total income in favour of men.
  • In this sample of 22,123 people, men’s average total income is $16,401 more than women’s.

We could also compare the medians (especially useful when dealing with income variables): SPSS gives the median total income of women in the sample as $23,000, while the median total income for men is $35,000 — a difference of medians of $12,000, again in favour of men.

To summarize, you can explore a potential association between a discrete and a continuo us variables of interest in two ways: 1) visually — by plotting and comparing boxplots, and 2) numerically, by inspecting the means (or medians) for the groups (i.e., the categories in the discrete variable being compared) and reporting their difference.

Keep in mind that we are not estimating anything at this point and are not claiming anything about the population: we are simply describing data based on a specific, actual sample. Figure 7.2 below shows a quick reference for interpreting boxplots. Figure 7.2 How to Interpret a Boxplot What Is Discrete Data This is how you can get boxplots like the ones in Figure 7.1 above:

From the Main Menu, select Graphs, then from the pull-down menu Legacy Dialogues, and finally Boxplot ; In the resulting Boxplot window select Simple and, keeping Summaries of groups of cases checked, click Define ; Select your continuous variable of interest from the list on the left and, using the appropriate arrow, move it into the Variable empty space on the right (at the top); Select your discrete variable of interest from the list on the left and, using the appropriate arrow, move it into the Category Axis empty space on the right (below the Variable ), then click OK ; Your boxplots will appear in the Output window. (Note that the graph will appear in its default SPSS colours and specifications. Double-clicking the chart will make a Chart Editor window appear. In the Chart Editor you can change, edit, and modify the appearance of your boxplots to your heart’s content.)

This is how you can get means, medians (or any descriptive statistic really) for different groups :

From the Main Menu, select Data and then from the pull-down menu, select Split File ; In the new window, select Compare groups, then find your discrete variable of interest from the left-hand side, and using the arrow, move it into the Groups Based on empty space; click OK, You would have just placed a filter on your data. From this point on (until you switch the filter off), everything you do in SPSS will be done for each separate group (this is indicated by a message “SORT CASES BY, SPLIT FILE LAYERED BY,” appearing in the Output window. Then, from the Main Menu, select Analyze, and then Frequencies, etc. to request any descriptive statistics you may like, e.g., the mean, the median, the standard deviation, etc. as discussed in the SPSS Tips in Chapter 3. Your output in the Output window will list the requested descriptives by the different groups (categories of the discrete variable). Once you are done with the comparisons, do not forget to switch the filter off (or your data file will remain split by groups): go again to Data in the Main Menu, select Split File and click Analyze all cases, do not create groups on the right-hand side; click OK. Your Output window will give a message of “SPLIT FILE OFF.” to indicate that the data is no longer split by group and it’s in its original condition.

Now let’s see how to “spot” and describe potential associations between two discrete variables.

Is income discrete or continuous?

1.5 Discrete and Continuous Variables I will introduce a final useful typology by which variables can be grouped: discrete and continuous. By definition, variables called discrete (note, not discreet!) have finite number of categories (i.e.,”space” between them, and nothing occupies that space), while variables called continuous have potentially infinite number of values (i.e., it’s possible that a value exists between any two given values, in smaller and smaller — infinite — number of “spaces” between any two the values, to infinity).

To make things easier to understand, and with more than a little risk of oversimplification, in a very broad sense you can think of nominal and ordinal variables as discrete and of interval/ratio variables as continuous, For example, hair colour, religious affiliation, and educational attainment (as measured in educational degrees) are all discrete: they have finite number of discrete categories.

On the other hand, age, income, or exam scores are all continuous: a number (value) can exist between any two given values, depending on how precise you want your measurement to be. To take age, for example, if two people report being 20 and 22, respectively, it’s obviously possible that another person in 21.

  1. However, we need not round to full years; between two people ages 20 and 21, a value of 21.5 (or 21 years and 6 months) is possible to exist.
  2. Further, between the ages of 21 years and 21 years and 6 months, we can have a value of 21 years and 3 months, and so on, until we are down to counting days, then counting hours, then counting minutes, then counting seconds, then milliseconds, then microseconds, then nanoseconds, etc.

The point is that, in theory, there is always a smaller number between any two numbers (which can be represented by the possibility of infinite number of digits after the decimal point). The same can be applied to income and exam scores too. In practice, however, things are different.

In sociological research (as with other similar disciplines), the data collected is empirically discrete, as the values collected are a finite number and are typically rounded to whole numbers: we don’t bother to measure age in anything but years, income in dollars (and not cents), etc. Still, w e usually call interval/ratio variables are continuous, because of the potential for infinite number of values.

At the same time, however, some ratio variables are truly discrete. Think, for example, about a measure called number of children of the respondent. Clearly, there is no possibility for an infinite number of values, just like with any “number of people”-type variable: people can only be counted in whole numbers, and the count is always finite.

  • All this is undoubtedly confusing, so here is a practical tip for applied research, and what you need to focus on.
  • Regardless if a variable is discrete or continuous in theory, in practice all variables you will encounter in real-life, actual datasets will be discrete.
  • What we do is treat some variables as discrete, and other variables as continuous for the purposes of statistical analysis,

The rule of thumb is to make the differentiation based on the number of categories/values: typically nominal and ordinal variables have relatively few categories so we treat them as discrete, while interval/ratio variables typically have relatively large number of values, so we treat them as continuous.

If, however, an ordinal variable has relatively large number of categories it may be treated as continuous, and, on the flip side, if an interval/ratio variable has relatively few values it may be treated as discrete. Generally, and assuming proper justification (i.e., a large number of categories/values), the decision to treat an ordinal variable as continuous or an interval/ratio variable as discrete remains a matter of the researcher’s discretion.

Finally, what is the magic number in the “relatively large number of categories/values” rule? This also depends, but from what I have seen in practice, the number is around 7-10 categories/values for most (i.e., if a variable has more categories/values that that it’s treated as continuous, and if it has fewer categories/values than that it is treated as discrete).

What type of data is discrete?

Discrete Data – The term discrete means distinct or separate. The discrete data contain the values that fall under integers or whole numbers. The total number of students in a class is an example of discrete data. These data can’t be broken into decimal or fraction values.

Is temperature discrete or continuous?

Climate Time Series: Random variables Random Variable (Continuous; Discrete) is a function which assigns a numerical value to all possible outcomes of an experiment. The values of random variables differ from one observation to the next in a manner described by their probability distribution.

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Ross, 1994] Discrete is a type of random variable which may take on only a limited set of integer values, such as 1,2,3,.,10. The list may be finite, or there may be an infinite number of values. A discrete random variable is to be contrasted with a continuous random variable. (Variables in Nominal and Ordinal Scales) Continuous is a type of random variable which may take any real value over an interval.

This is contrasted with a discrete random variable, which may take only a limited set of values. (Variables on Interval and Ratio scales) Climate Variables (temperature, precipitation, wind, evaporation, etc.) fit the definition of random variables as each observation is different from the next, and there is a function that describes quantity of the variable for each observation.

For example, temperature has a continuous character (thermometer can take measurements anytime), whose amount at a given time depends (is a function of) on many factors. Temperature measurement could be classified as an example of a continuous random variable that is measured on interval scale. Character of precipitation is another example of a random variable – either as a discrete variable: precipitation status (nominal scale: days with precipitation, trace of precipitation, no precipitation) at a given time or as continuous random variable bounded at zero when treated as amount of precipitation (variable at a ratio scale).

REFERENCES: Ross Sh., 1994. A First Course in Probability, 4 ed., Macmillan College Publishing Company, Inc., pp.126-176

Climate Time Series: Random variables

Is height discrete or continuous?

1331.0 – Statistics – A Powerful Edge!, 1996 ORGANISING DATA After collection and processing, data need to be organised to produce useful information. It helps to be familiar with some definitions when organising data. This section outlines those definitions and provides some simple techniques for organising and presenting data.

  • Definition The word variable is often used in the study of statistics and so it is important to understand its meaning.
  • Height, age, amount of income, country of birth, grades obtained at school and type of housing are examples of variables.

Variables may be classified into various categories, some of which are outlined in the following pages. NOMINAL VARIABLES

A nominal (also called categorical) variable is one that describes a name or category. EXAMPLE 1. The method of travel to work by people in Darwin at the time of the 1996 Census was:

Method of travel to work Number of people
WALKED 1,703
BUS 1,335
TAXI 284

In this case the variable ‘method of travel to work’ is nominal because it describes a name. NUMERIC VARIABLE A numeric variable is one that describes a numerically measured value. However, not all variables described by numbers are numeric. For example, the age of a person is a numeric variable, but their year of birth, despite being a number, is a nominal variable.

Numeric variables may be either continuous or discrete: CONTINUOUS VARIABLE A variable is said to be continuous if it can take any value within a certain range, Examples of continuous variables may be distance, age or temperature. The measurement of a continuous variable is restricted by the methods used, or by the accuracy of the measuring instruments.

For example, the height of a student is a continuous variable because a student may be 1.6321748755. metres tall. However, when the height of a person is measured, it is usually only measured to the nearest centimetre. Thus, this student’s height would be recorded as 1.63m.

Note that continuous variables are usually grouped using class intervals (explained shortly). They are grouped to make them easier to handle as part of the general process of organising data into information. DISCRETE VARIABLE Any variable that is not continuous is discrete. A discrete variable can only take a finite number of values within a certain range,

An example of a discrete variable would be a score given by a judge to a gymnast in competition: the range is 0-10 and the score is always given to one decimal place. Discrete variables may also be grouped. Again, this is done to make them easier to handle.

Behaviour Number of Students
Excellent 5
Very Good 12
Good 10
Bad 2
Very bad 1

In this case the variable ‘behaviour’ is nominal and ordinal, : 1331.0 – Statistics – A Powerful Edge!, 1996

Is 50 years old discrete or continuous?

Answer and Explanation: Age is commonly measured in years, which would make it a discrete variable.

Is hours discrete or continuous?

$\begingroup$ It can be viewed both as discrete and as continuous. In fact we have discrete-time and continous-time models. It depends how did you record the time, e.g. if you count days, or record hours rounded to the nearest hour then it is rather discrete; when you record days, hours and minutes of something happening, then it is closer to continuous.

  • With real-life data we can never be infinitely precise with our measurements, so one could argue that every measurement is discrete given the finite precision.
  • Finally, models designed for continuous variables can in many cases be used with discrete variables (but using models for discrete data with continuous variables is rather a bad idea).

answered Jun 27, 2016 at 10:37 What Is Discrete Data Tim ♦ Tim 134k 24 gold badges 246 silver badges 479 bronze badges $\endgroup$ 1

$\begingroup$ +1 But it is more than simple measurement: there are ontological theories of discrete time (e.g. school years, fiscal years, work weeks, etc.), and ontological theories of continuous time (e.g. half-life of radioactive decay). $\endgroup$ Mar 7, 2017 at 1:51

Is Speed a continuous or discrete?

Yes, speed is a common continuous variable, and the value is chosen by a random process. We know it is continuous because there is always another possible value between any two speed values. It would not be possible to count all of the possible speeds that could be.

Is month discrete or continuous?

There are #12# months in the Gregorian calendar. Twelve is a counting number. #12 in NN# Since the counting numbers ( #NN# ) are countable, they deal with the discrete. Discrete is another way of saying not-continuous. Therefore, the calendar months would be a non-continuous random variable.

Is shoe size discrete or continuous?

Shoe size is a Discrete variable.E.g.5, 5½, 6, 6½ etc. Not in between. Temperature is a continuous variable.

What makes something discrete?

From working with statistics, we know that data can be numerical ( quantitative ) or descriptive ( qualitative ). When data is numerical, it can also be discrete or continuous,

Continuous Definition: A set of data is said to be continuous if the values belonging to the set can take on ANY value within a finite or infinite interval. Definition: A set of data is said to be discrete if the values belonging to the set are distinct and separate (unconnected values). Examples: • The height of a horse (could be any value within the range of horse heights). • Time to complete a task (which could be measured to fractions of seconds). • The outdoor temperature at noon (any value within possible temperatures ranges.) • The speed of a car on Route 3 (assuming legal speed limits). Examples: • The number of people in your class (no fractional parts of a person). • The number of TV sets in a home (no fractional parts of a TV set). • The number of puppies in a liter (no fractional puppies). • The number of questions on a math test (no incomplete questions). NOTE: Continuous data usually requires a measuring device. (Ruler, stop watch, thermometer, speedometer, etc.) NOTE: Discrete data is counted, The description of the task is usually preceded by the words “number of.”, Function: In the graph of a continuous function, the points are connected with a continuous line, since every point has meaning to the original problem. Function: In the graph of a discrete function, only separate, distinct points are plotted, and only these points have meaning to the original problem. Graph: You can draw a continuous function without lifting your pencil from your paper. Graph: A discrete graph is a series of unconnected points (a scatter plot). Domain: a set of input values consisting of all numbers in an interval. Domain: a set of input values consisting of only certain numbers in an interval. In Plain English: A continuous function allows the x -values to be ANY points in the interval, including fractions, decimals, and irrational values. In Plain English: A discrete function allows the x -values to be only certain points in the interval, usually only integers or whole numbers. Why do we care? When graphing a function, especially one related to a real-world situation, it is important to choose an appropriate domain ( x -values) for the graph. For example, if a function represents the number of people left on an island at the end of each week in the Survivor Game, an appropriate domain would be positive integers. Hopefully, half of a person is not an appropriate answer for any of the weeks. The graph of the people remaining on the island would be a discrete graph, not a continuous graph.

Is IQ discrete or continuous?

Why IQ test “discrete scores” are said to have “Normal shape” IQ test scores don’t increment in decimals, but in whole numbers (e.g., 138, 140, 150 etc.). In other words, IQ tests only provide discrete scores (No body can get an IQ score of $115.568.$).

If so, how is it that we say that population IQ test scores has a $Normal~Shape$ with mean of $100$ and sigma of $15$, normality doesn’t apply to discrete variables? More generally in human research, no one uses thermometer-type (precise and graduated) measurement tools, instead they often use measurement tools that produce discrete scores.

How assumptions of normality to perform statistical tests such as t-test can apply then? : Why IQ test “discrete scores” are said to have “Normal shape”

What is not discrete data?

Continuous data — it’s all about accuracy – Continuous data is considered the complete opposite of discrete data. It’s the type of numerical data that refers to the unspecified number of possible measurements between two presumed points. The numbers of continuous data are not always clean and integers, as they are usually collected from very precise measurements.

  • Measuring a particular subject is allowing for creating a defined range to collect more data.
  • Variables in continuous data sets often carry decimal points, with the number stretching out as far as possible.
  • Typically, it changes over time.
  • It can have completely different values at different time intervals, which might not always be whole numbers.

Here are some examples:

The weather temperature;The wind speed;The weight of the kids;

Continuous data can be measured by using specific tools and displayed in line graphs, skews, histograms.

What is discrete with example?

What is Discrete Data? Definition, Examples, and Explanation – If you have quantitative data, like a number of workers in a company, could you divide every one of the workers into 2 parts? The answer is absolutely NOT. Because the number of workers is discrete data.

Let’s define it: Discrete data is a count that involves integers. Only a limited number of values is possible. The discrete values cannot be subdivided into parts. For example, the number of children in a school is discrete data. You can count whole individuals. You can’t count 1.5 kids. So, discrete data can take only certain values.

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The data variables cannot be divided into smaller parts. How to display graphically discrete data? We can display discrete data by bar graphs. Stem-and-leaf-plot and pie chart are great for displaying discrete data too. Discrete data key characteristics:

You can count the data. It is usually units counted in whole numbers. The values cannot be divided into smaller pieces and add additional meaning. You cannot measure the data. By nature, discrete data cannot be measured at all. For example, you can measure your weight with the help of a scale. So, your weight is not a discrete data. It has a limited number of possible values e.g. days of the month.Discrete data is graphically displayed by a bar graph,

Discrete data may be also ordinal or nominal data (see our post nominal vs ordinal data ). When the values of the discrete data fit into one of many categories and there is an order or rank to the values, we have ordinal discrete data. For example, the first, second and third person in a competition.

The number of students in a class.The number of workers in a company.The number of parts damaged during transportation.Shoe sizes.Number of languages an individual speaks.The number of home runs in a baseball game.The number of test questions you answered correctly.Instruments in a shelf.The number of siblings a randomly selected individual has.

Why is it called discrete?

Introduction to Discrete Structures – Whats and Whys – What is Discrete Mathematics ? Discrete mathematics is mathematics that deals with discrete objects. Discrete objects are those which are separated from (not connected to/distinct from) each other.

Integers (aka whole numbers), rational numbers (ones that can be expressed as the quotient of two integers), automobiles, houses, people etc. are all discrete objects. On the other hand real numbers which include irrational as well as rational numbers are not discrete. As you know between any two different real numbers there is another real number different from either of them.

So they are packed without any gaps and can not be separated from their immediate neighbors. In that sense they are not discrete. In this course we will be concerned with objects such as integers, propositions, sets, relations and functions, which are all discrete.

We are going to learn concepts associated with them, their properties, and relationships among them among others. Why Discrete Mathematics ? Let us first see why we want to be interested in the formal/theoretical approaches in computer science. Some of the major reasons that we adopt formal approaches are 1) we can handle infinity or large quantity and indefiniteness with them, and 2) results from formal approaches are reusable.

As an example, let us consider a simple problem of investment. Suppose that we invest $1,000 every year with expected return of 10% a year. How much are we going to have after 3 years, 5 years, or 10 years ? The most naive way to find that out would be the brute force calculation.

  • Let us see what happens to $1,000 invested at the beginning of each year for three years.
  • First let us consider the $1,000 invested at the beginning of the first year.
  • After one year it produces a return of $100.
  • Thus at the beginning of the second year, $1,100, which is equal to $1,000 * ( 1 + 0.1 ), is invested.

This $1,100 produces $110 at the end of the second year. Thus at the beginning of the third year we have $1,210, which is equal to $1,000 * ( 1 + 0.1 )( 1 + 0.1 ), or $1,000 * ( 1 + 0.1 ) 2, After the third year this gives us $1,000 * ( 1 + 0.1 ) 3, Similarly we can see that the $1,000 invested at the beginning of the second year produces $1,000 * ( 1 + 0.1 ) 2 at the end of the third year, and the $1,000 invested at the beginning of the third year becomes $1,000 * ( 1 + 0.1 ).

Thus the total principal and return after three years is $1,000 * ( 1 + 0.1 ) + $1,000 * ( 1 + 0.1 ) 2 + $1,000 * ( 1 + 0.1 ) 3, which is equal to $3,641. One can similarly calculate the principal and return for 5 years and for 10 years. It is, however, a long tedious calculation even with calculators.

Further, what if you want to know the principal and return for some different returns than 10%, or different periods of time such as 15 years ? You would have to do all these calculations all over again. We can avoid these tedious calculations considerably by noting the similarities in these problems and solving them in a more general way.

Since all these problems ask for the result of invesing a certain amount every year for certain number of years with a certain expected annual return, we use variables, say A, R and n, to represent the principal newly invested every year, the return ratio, and the number of years invested, respectively.

With these symbols, the principal and return after n years, denoted by S, can be expressed as S = A (1 + R ) + A (1 + R ) 2 +, + A (1 + R ) n, As well known, this S can be put into a more compact form by first computing S – (1 + R ) S as S = A ( (1 + R ) n + 1 – (1 + R ) ) / R,

Once we have it in this compact form, it is fairly easy to compute S for different values of A, R and n, though one still has to compute (1 + R ) n + 1, This simple formula represents infinitely many cases involving all different values of A, R and n, The derivation of this formula, however, involves another problem.

When computing the compact form for S, S – (1 + R ) S was computed using S = A (1 + R ) + A (1 + R ) 2 +, + A (1 + R ) n, While this argument seems rigorous enough, in fact practically it is a good enough argument, when one wishes to be very rigorous, the ellipsis,

In the sum for S is not considered precise. You are expected to interpret it in a certain specific way. But it can be interpreted in a number of different ways. In fact it can mean anything. Thus if one wants to be rigorous, and absolutely sure about the correctness of the formula, one needs some other way of verifying it than using the ellipsis.

Since one needs to verify it for infinitely many cases (infinitely many values of A, R and n ), some kind of formal approach, abstracted away from actual numbers, is required. Suppose now that somehow we have formally verified the formula successfully and we are absolutely sure that it is correct.

  1. It is a good idea to write a computer program to compute that S, especially with (1 + R ) n + 1 to be computed.
  2. Suppose again that we have written a program to compute S,
  3. How can we know that the program is correct ? As we know, there are infinitely many possible input values (that is, values of A, R and n ).

Obviously we can not test it for infinitely many cases. Thus we must take some formal approach. Related to the problem of correctness of computer programs, there is the well known “Halting Problem”. This problem, if put into the context of program correctness, asks whether or not a given computer program stops on a given input after a finite amount of time.

This problem is known to be unsolvable by computers. That is, no one can write a computer program to answer that question. It is known to be unsolvable. But, how can we tell it is unsolvable ?. How can we tell that such a program can not be written ? You can not try all possible solution methods and see they all fail.

You can not think of all (candidate) methods to solve the Halting Problem. Thus you need some kind of formal approaches here to avoid dealing with a extremely large number (if not infinite) of possibilities. Discrete mathematics is the foundation for the formal approaches.

  • It discusses languages used in mathematical reasoning, basic concepts, and their properties and relationships among them.
  • Though there is no time to cover them in this course, discrete mathematics is also concerned with techniques to solve certain types of problems such as how to count or enumerate quantities.

The kind of counting problems includes: How many routes exist from point A to point B in a computer network ? How much execution time is required to sort a list of integers in increasing order ? What is the probability of winning a lottery ? What is the shortest path from point A to point B in a computer network ? etc.

  • The subjects covered in this course include propositional logic, predicate logic, sets, relations, and functions, in particular growth of function.
  • The first subject is logic.
  • It is covered in Chapter 1 of the textbook.
  • It is a language that captures the essence of our reasoning, and correct reasoning must follow the rules of this language.

We start with logic of sentences called propositional logic, and study elements of logic, (logical) relationships between propositions, and reasoning. Then we learn a little more powerful logic called predicate logic. It allows us to reason with statements involving variables among others.

  1. In Chapter 1 we also study sets, relations between sets, and operations on sets.
  2. Just about everything is described based on sets, when rigor is required.
  3. It is the basis of every theory in computer science and mathematics.
  4. In Chapter 3 we learn mathematical reasoning, in particular recursive definitions and mathematical induction.

There are sets, operations and functions that can be defined precisely by recursive definitions. Properties of those recursively defined objects can be established rigorously using proof by induction. Then in Chapter 6 we study relations. They are an abstraction of relations we are familiar with in everyday life such as husband-wife relation, parent-child relation and ownership relation.

They are also one of the key concepts in the discussion of many subjects on computer and computation. For example, a database is viewed as a set of relations and database query languages are constructed based on operations on relations and sets. Graphs are also covered briefly here. They are an example of discrete structures and they are one of the most useful models for computer scientists and engineers in solving problems.

More in-depth coverage of graph can be found in Chapter 7. Finally back in Chapter 1 we study functions and their asymptotic behaviors. Functions are a special type of relation and basically the same kind of concept as the ones we see in calculus. However, function is one of the most important concepts in the discussion of many subjects on computer and computation such as data structures, database, formal languages and automata, and analysis of algorithms which is briefly covered in Chapter 2.

What does it mean to be discrete?

Discreet means on the down low, under the radar, careful; but discrete means ‘individual’ or ‘detached.’ They come from the same ultimate source, the Latin discrētus, for ‘separated or distinct,’ but discreet has taken its own advice and quietly gone its separate way.