গণসংখ্যা নিবেশন সারণি

I'm sure you know what a table is already, but what is frequency? Frequency is how often something occurs, i.e., the number of times something occurs. Frequency is a branch of statistics.

For example, if you had a data set like $1,1,1,2,2$, we would say $1$ has a frequency of $3$ since it appears three times.

A frequency table is a table of the data with their respective frequencies. It shares similarities to histograms and stem and leaf plots.

With our data set $1,1,1,2,2$, the table would be: $$\begin{array}{|c|c|} \hline \text{Number}&\text{Frequency}\\ \hline 1&3\\ 2&2\\ \hline \end{array}$$

Intervals

When we have larger data sets or larger ranges, we can use intervals to construct the frequency table.

Let's create a frequency table for the following data set.

$$30,13,39,28,18,36,22,36,42,43,36,29,34,40,21$$

Since this is kind of confusing to look at, let's put the data in ascending order first.

$$13,18,21,22,28,29,30,34,36,36,36,39,40,42,43$$

So, instead of counting the frequencies for $13, 18, 21,$ and so on, we can group them into intervals like $10-19$. Then, we just find the frequency for the intervals.

Filling in the frequency table, we get:

$$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 10-19&2\\ 20-29&4\\ 30-39&6\\ 40-49&3\\ \hline \end{array}$$

How do we know how many intervals to use and what their sizes should be? Unfortunately, there isn't really a definitive answer. Intervals shouldn't be too big and shouldn't be too small.

Here's another frequency table of the data but with different intervals.

$$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 0-15&1\\ 16-30&6\\ 31-45&8\\ \hline \end{array}$$

Both of these are valid.

Intervals with Decimals

One important aspect is what we do with intervals when the data isn't only integers. Suppose we have the following data:

$$1.13,1.9,1.999,3,3.1,4,5,6.2,6.5,6.56,7.8,7.801,8.99$$

You might set the intervals for the table as follows:

$$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 1-3&\\ 4-6&\\ 7-9&\\ \hline \end{array}$$

But where would a number like $3.1$ fall for this?

When dealing with decimals, we can use stricter intervals.

$$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline [1-3)&\\ [3-6)&\\ [6-9)&\\ \hline \end{array}$$

Recall that the bracket means we include that number and the parenthesis means we go up to but don't include that number. So the first interval includes $1$ and all the numbers that go up to $3$, but it doesn't include $3$.

When making the frequency table, we don't include the brackets or the parentheses. It's just agreed upon that the left value in inclusive and the right value is exclusive.

$$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 1-3&\\ 3-6&\\ 6-9&\\ \hline \end{array}$$

If we fill in the frequencies, we get:

$$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 1-3&3\\ 3-6&4\\ 6-9&6\\ \hline \end{array}$$

Reading the Table

Other than constructing a frequency table, you will also be asked to gather information from the table.

Suppose we have the following frequency table: $$\begin{array}{|c|c|} \hline \text{Interval}&\text{Frequency}\\ \hline 1-5&6\\ 6-10&3\\ 11-15&8\\ \hline \end{array}$$ How many data points are there?

We can always figure out how many data points there are by adding all the frequencies together. $$6+3+8=17$$

What fraction of the data is between $6-10$?

To find this, we can take the frequency of the interval we want, $6-10$, and place it over the total frequency. $$\frac{3}{6+3+8}=\frac{3}{17}$$

How many data points are below $11$?

Just add up the frequencies of the intervals that are less than $11$. These would be $1-5$ and $6-10$. $$6+3=9$$

Finally, there is some data that we can't find when using a frequency table. Frequency tables are a way to quickly generalize data. They are easier to read but the trade off is they can be less accurate.

For example, if you were asked how many $3$'s were in the data given the above frequency table, we couldn't answer. We know the frequency for the values in the interval $1-5$, but we don't know how many were exactly $3$.

Oh How the Tables Have Turned...Into Histograms

Once you master making frequency tables, we can then turn the tables into a type of bar graph. These graphs are called histograms, which we'll learn next.

(Video) Frequency Tables and Histograms

by Anywhere Math

This video by Anywhere Math goes over how to construct a frequency table and then how to turn that into a histogram.

The frequency of something is just how often that something occurs. If you rolled a dice and got a $1$ four times, then the frequency of rolling a $1$ is $4$.

A frequency table is a table that shows the frequency of data values in same sized intervals. "A table that organizes your data into intervals."

Example

The first example he looks at is turning the data of shoe sizes in a class into a frequency table. The data he has is: $$4,4.5,5,5,6,6.5,7,7,7,8,9.5,10$$

Good Practice Tip: Put your data in ascending order.

We start by making the outline of our frequency table. This is easy. There are only two columns.

$$\begin{array}{|c|c|} \hline \text{Sizes}&\text{Frequency}\\ \hline \\ \\ \\ \\ \hline \end{array}$$

Next, we need to make intervals for our shoe sizes. There are a number of ways to break up these shoe sizes into intervals. We could do sizes from $0$ to $6$ and then from $6$ to $12$, but two intervals usually isn't good.

Good Practice Tip: Try to use $4$ to $5$ intervals. Larger data sets might require more intervals.

So, let's put our data into $4$ intervals: $4-6, 6-8, 8-10, 10-12$.

$$\begin{array}{|c|c|} \hline \text{Sizes}&\text{Frequency}\\ \hline [4,6)&\\ [6,8)&\\ [8,10)&\\ [10,12]&\\ \hline \end{array}$$

Good Practice Tip: Start the new interval with the number you left off with. This allows decimals that would appear between intervals to have a place.

Next, we look at our data, and tally up how many go into each interval.

$$\color{#8925AE}{4},\color{#8925AE}{4.5},\color{#8925AE}{5},\color{#8925AE}{5},6,6.5,7,7,7,8,9.5,10$$

These four definitely go into the interval of $4-6$. But this raises a question, what about $6$ itself? Does it go into the interval $4-6$ or $6-8$? Standard practice says to include the number if the interval starts with it. So $6$ wouldn't fall into the interval $4-6$ but it would for $6-8$.

$$\begin{array}{|c|c|} \hline \text{Sizes}&\text{Frequency}\\ \hline 4-6&4\\ 6-8&\\ 8-10&\\ 10-12&\\ \hline \end{array}$$

Do the same for the rest of the data.

$$\begin{array}{|c|c|} \hline \text{Sizes}&\text{Frequency}\\ \hline 4-6&4\\ 6-8&5\\ 8-10&2\\ 10-12&1\\ \hline \end{array}$$

In the next explanation, we will look at turning this table into a histogram.

Frequency and Frequency Tables

In algebra, when we talk about the frequency of a value, we're talking about how many times it occurs.

For example, if there's a class full of students that just took a math test and 6 of them scored an $85\%$, the frequency of that score $(85\%)$ is $6$. For now, think of the frequency as *just a count *of how often a value occurs.

Frequency tables are an easy way to organize a bunch of messy data. You can represent frequencies in a table as plain old numbers or as tallies.

For example, imagine that you run a restaurant that serves chili. You've asked your waiters to keep track of how many of each type of chili they serve. Here are your data:

To organize this, set up a frequency table. You can use numbers, tallies, or both. It's up to you (or whoever is writing the math problem).

We can take this a step further by looking at frequency distributions, cumulative frequency tables, and two-way frequency tables.

Frequency Tables

Frequency tables are a great way of representing data that have multiples of the same items.

Say we want to create a table to represent how many different forms of media we have in our room. We have $4$ books, $6$ magazines, and $2$ TV show DVDs. How would we represent this in a clear and easy-to-read way? Well, we'd construct a frequency table!

By representing the data in this manner, we're able to easily interpret the data. Frequency tables also make it easier to construct a histogram.