Summarizing Categorical Data: Frequency and Relative Frequency Tables

okay so let's begin with section 2.1 uh which deals entirely with categorical data right so when we work

with categorical data remember that's any data that is names labels addresses descriptors

right so um there may be a natural way to sort um so so you can collect them at the

ordinal level or maybe we don't even have that we just have the nominal level of measurement for for these data what i

have here is a simulated sample of categorical data

namely um these are descriptors of students at hypothetical students at arc

um who are either first year students second year third year fourth year um or they are

ready to transfer meaning they're going to transfer next semester so you'll notice that there are

four rows four columns so that means there's 16 data values here so i'm just going to

write that down so our sample size here because this is a sample is 16. that'll be important later

whenever we are going to be ready for organizing these data um

but you'll notice that if we want to sort them sorting in this case meaning

so to sort data is to organize them by some um organizing principle so you could say

okay put the first years together first and the second years so we'll decide that

um or it might be given to us so first year second year these are those the unique kinds of labels that we can

have right if we were talking about um gender or racial identity we would have

unique identifiers or variables that these data values then follow under

third year fourth year and transfer there might be more but these are all the ones that we have

in our in our data set so now that we've we have a sorting

principle right we would then actually sort them right this

you can use technology to help you so for instance if you're using um let me use a different color here

so for tech right you might be using excel to the microsoft office tool if you put all of these in a

column um there's a tool that allows you to sort alphabetically or

um in other organizing principles right by hand which isn't really tech but you could sort by hand if you have a

small enough data set and there are other tools available available to us

if you put them directly into ti-84 for instance if you have um if you wanted to but it's not really

preferenced here for categorical data it would sort it for you once you go to use tables

but anyway so once you've sorted your data right um we're going to sort them into

piles which we kind of are doing here by sorting them into each distinct variable which actually partitions our

data already but organizing this way isn't enough what we usually want to do is we want to

summarize our data right so data summarization refers to using

condensing our raw data this is our raw data raw categorical data we want to condense

our data in some way that still maintains the level of measurement

so the ways that we can do this one common way is to use a table

and the way that we can use the table is we can count how many of each distinct kind of label there are

so let me illustrate and then we'll get into what each one means

so i'm setting up two columns here for my table one of them is labeled category and the

other one is labeled count slash frequency count and frequency are interchangeable

okay but what these this means is for each one of our distinct labels right so first second

third i don't have enough room here but so i'll finish this on on a separate sheet um

the idea is for each one of these unique identifiers or labels for our categorical data

we have to count how many times each one occurs in our data set so just for the sake of illustration

i'll i'll do this properly on another sheet here but how many first-year students are

there represented in our data set so i count one two three

four just 4. so we put a 4 there corresponding to the first year students

for second year students right how many are there there's 1 2 3 4

five how many third year students okay one two and so on until we finished

going through our entire data set you'll notice then that as we go on each one of these labels should get

counted once exactly once so properly though what we're doing is we're setting up a

frequency distribution for categorical data that's what that

table is called so we have a category or what we call class

column and we have a count or what we call frequency column

now to finish up that example from earlier i'm going to use class and frequency as

our we're going to be using these these uh terms much more often than category and

count but category is helpful in this case because it helps us remember that we're

actually working with categorical data so remember those first years second years

third years fourth years and transfers and we already said we found four first years five second years and two

second years four five two and we go back and we count

how many fourth years and how many transfers i count one fourth year and one

two three four transfers okay so that's our frequency distribution and you'll know

that you did this correctly if when you add up the frequencies you should get the sample size back

so if we add it up right so that's 9 11 10 no 9 11

12 16. that is our that is our sample size so we did this correctly

okay and what we might write notationally we might say that the sum this notation is called

sum you might want to review summation notation in the appendix for each of our classes i'm going to use

a subscript k the sum of the frequencies which we denote by f sub k

so these are just labels right we're saying we're adding up all the frequencies

that has to add up to the sample size so that's how we know that the frequency table works

this frequency distribution and that's not the only way that we can organize our

categorical data we not so frequency is helpful in we're actually sorting everything and compressing it

into this table but unless we're very clear about what the sample size is or

if we have a good sense of relative size we might need to do something better

we might use something called relative frequency and what that is is we take each

frequency as a proportion or a ratio out of the total sample size

so remember this is sample size i'm going to use words here

and what this does is this rescales our table so that we understand it in terms of proportions

percentages ratios right so when you're using relative frequency

you probably want to round to the same precision so typically we'll round to no more than

four places after the decimal you can do more than that but i don't recommend it

so again our i'll set up our frequency distribution again but i'm going to add

a column when we do the relative frequency distributions we won't have the middle column

necessarily but you can if you want if it helps you set it up so now i'm going to have three

columns here so for my first year students second year students third year

fourth year and my transfer students okay and then so we said there were four five

two one and four four five two one and four until we'll bring that down again and to

calculate each relative frequency we're going to compute that frequency divided by remember what

was the sample size we said n equals 16. so 4 divided by 16 and you can leave it as a fraction or

if you want to represent it as a decimal number then you should actually compute that

now 4 16 that's 0.25 because that's a quarter but the rest we probably will need some

help figuring that out so pull out your calculator whatever you have

um i've had i am using my phone calculator because we don't need much more than that

now 5 16 that's 0.3125 2 16. is 0.125 and 1

16 is 0.0625 okay so each of these we're actually representing

as a so remember no more than four places after the decimal i added the zeros for just for the sake

of um clarity and so that it all looks the same but

so as we're as you're working with the relative frequencies remember there was that rule when we

worked with just the frequencies that they have to add up to the sample size

in this case because we divided each of them by the sample size they should add up to one right so you

can verify using your calculator as i'm doing right now and sometimes you'll be a little bit off

just because especially if you're rounding if you round your decimals then

you might be off by .01 or something very small but if you're not rounding then it

should be precise it should equal exactly one and so with that again that that

rule that tells you that you did this correctly for relative frequency we would say that the sum

remember this symbol sigma represents sum over all the classes

of the relative frequencies equals 1. so that's how you know you've done it correctly

and this will come back again in a later uh unit but you will you will be able to use this

not only with categorical data but also with quantitative data so we'll see that in a later video