1.2: Understanding and Interpreting Frequency Tables and Histograms
Do you understand histograms? Did you know that histograms can be a useful way of explaining data?
Jessie works at an ice cream stand. For one hour she recorded the ages of the children who came with their parents to buy ice cream. The histogram above shows this information. Looking at the histogram, can you determine how many persons of each age came to the ice cream stand?
Pay attention and you will know how to answer this question by the end of the Concept.
Guidance
To understand what a frequency table is, let’s first look at the words themselves. The word frequency refers to how often something occurs. A table is a way of organizing information using columns. Therefore, a frequency table is way of summarizing data by depicting the number of times a data value occurs. To show this, a frequency table organizes the information into a table with three separate columns.
How do we create a frequency table?
First, you need to make a table with three separate columns.
One column is designated for intervals. The number of intervals is determined by the range of data values. Intervals are equal in size and do not overlap. If the range in data values is close together, then the intervals will be small. If the range in data values is spread apart, then the intervals will be larger. It is important that the range of values in each interval is equal and that the values do not overlap from one interval to the next.
Another column is created for tallied results. This is where you tally the number of times you see a data value from each interval. This is where you will see tally marks or lines that record the number of times a data value occurs.
In the last column, add the tally marks to determine the frequency results.
Let’s look at creating a frequency table. Create a frequency table to display the data below.
Step 1: Make a table with three separate columns.
- Intervals
- Tallied results
- Frequency results
Since the range in data values is not that great, intervals will be in groups of five.
Step 2: Looking at the data, tally the number of times a data value occurs.
Step 3: Add the tally marks to record the frequency.
Take a few minutes to write down the steps for creating a frequency table.
Thinking about the frequency that an event occurs can help you to understand and predict certain trends. Think about how useful the trend of grades could be if you were a teacher thinking about a student’s progress.
We can also create a histogram to display data. Histograms and bar graphs are often confused, but they are different. Let’s look at how.
A histogram shows the frequency of data values on a graph. Like a frequency table, data is grouped in intervals of equal size that do not overlap. Like a bar graph, the height of each bar depicts the frequency of the data values. However, on a histogram the vertical columns have no space in between each other.
Create a histogram to display the information on the frequency table.
Here are the steps for creating a histogram from data organized in a frequency table.
Step 1: Draw the horizontal and vertical axis.
Step 2: Give the graph the title “Frequency Table Data.”
Step 3: Label the horizontal axis “Hours.” List the intervals across the horizontal axis.
Step 4: Label the vertical axis “Frequency.” Since the range in frequencies is not that great, label the axis by ones.
Step 5: For each interval on the horizontal access, draw a vertical column to the appropriate frequency value. On a histogram, there is no space in between vertical columns.
Looking at the histogram, you can see that data values between thirty-six and forty were most frequent. Data values between forty-one and forty-five and forty-six and fifty occurred an equal number of times.
Use this histogram of scores earned on math exam to answer the following questions.
Example A
Where did the majority of the scores fall?
Solution: Between eighty-six and ninety- five percent
Example B
What fraction of the students earned between seventy-six and eighty-five percent?
Solution: One-fourth
Example C
Which scores were in the minority?
Solution: Between ninety-six and one hundred and five percent
Now back to the dilemma from the beginning of the Concept. Here is the histogram once again.
Let's use it to answer these questions.
What was the most popular age group at the ice cream stand?
There were seven, seven year old children.
How many one year old children came?
None
How many eight year old children came?
Three
How many ten year old children came?
None
Notice that you could write many questions and answers using this histogram. The histogram provides a wonderful visual display of the data.
Vocabulary
- Data
- information that has been collected regarding an occurrence or an event.
- Bar Graph
- a graph that uses columns to compare quantities or amounts.
- Frequency Table
- a display that summarizes data by depicting the number of times that a data value occurs.
- Histogram
- a display that shows the frequency of data values on a graph.
Guided Practice
Here is one for you to try on your own.
The data values below depict student scores (out of 100%) on a recent math exam. Organize the data into a frequency table.
Solution
Step 1: Make a table with three separate columns.
- Intervals
- Tallied results
- Frequency results
Since the range in data is big (thirty-one), intervals will be in groups of ten.
Step 2: Looking at the data, tally the number of times a data value occurs.
Step 3: Add the tally marks to record the frequency.
Now our work is complete.
Video Review
Practice
Directions: Use the frequency table to answer the following questions. This frequency table shows scores from an history exam.
Score (%) | Tally | Frequency |
---|---|---|
50-60 | 4 | |
60-70 | 6 | |
70-80 | 11 | |
80-90 | 8 | |
90-100 | 4 |
1. How many students total took the test?
2. How many students scored between 70% and 80%?
3. What fraction of the students scored between 70% and 80%?
4. What percent of the students would that be?
5. How many students scored between 90% and 100%?
6. What fraction of the students scored between 90% and 100%?
7. If failing is below 60%, how many students did not pass the test?
8. True or false. The same number of students received the highest scores as did not pass the test.
Directions: Use this histogram on siblings to answer the following questions.
9. How many people surveyed have two siblings?
10. How many people surveyed have three siblings?
11. How many people are only children?
12. How many people have ten siblings?
13. How many people combined have four or five siblings?
14. How many people have only one sibling?
15. How many people have nine siblings?
bar chart
A bar chart is a graphic display of categorical variables that uses bars to represent the frequency of the count in each category.bar graph
A bar graph is a plot made of bars whose heights (vertical bars) or lengths (horizontal bars) represent the frequencies of each category, with space between each bar.bell curve
A normal distribution curve is also known as a bell curve.bell shaped
A bell shaped histogram is a histogram with a prominent ‘mound’ in the center and similar tapering to the left and right.binning
Binning involves separating your data separated into separate classes or categories.bins
Bins are groups of data plotted on the x-axis.class limits
Class limits are, collectively, the upper and lower limit of an interval.class mark
A class mark is the middle value, or average of the class limits.Data
Data is information that has been collected to represent real life situations, usually in number form.extreme outliers
Extreme outliers include points more than 3 times the middle half of your data. .frequency density
The vertical axis of a histogram is labelled frequency density.frequency distribution table
A frequency distribution table lists the data values, as well as the number of times each value appears in the data set.frequency polygon
A frequency polygon is a graph constructed by using lines to join the midpoints of each interval, or bin.Frequency table
A frequency table is a table that summarizes a data set by stating the number of times each value occurs within the data set.Histogram
A histogram is a display that indicates the frequency of specified ranges of continuous data values on a graph in the form of immediately adjacent bars.Interval
An interval is a range of data in a data set.left-skewed distribution
A left-skewed distribution has a peak to the right of the distribution and data values that taper off to the left.mild outliers
Mild outliers include data points that are more than 1.5 times the middle half of your data above the upper, or below the lower, quartiles.multimodal
When a set of data has more than 2 values that occur with the same greatest frequency, the set is called multimodal .normal distributed
If data is normally distributed, the data set creates a symmetric histogram that looks like a bell.Outlier
In statistics, an outlier is a data value that is far from other data values.Range
The range of a data set is the difference between the smallest value and the greatest value in the data set.relative cumulative frequency plot (ogive plot)
A relative cumulative frequency plot, or ogive plot, shows how the data accumulate across the different values of the variable.relative frequency histogram
A relative cumulative frequency histogram is a histogram except the vertical bars as the relative cumulative frequencies.right-skewed distribution
A right-skewed distribution has a peak to the left of the distribution and data values that taper off to the right.shape
The shape of a histogram can lead to valuable conclusions about the trend(s) of the data.skewed
As with the horizontal skewing of a histogram, stem plots with a obvious skew toward one end or the other tend to indicate an increased number of outliers either lesser than or greater than the mode.symmetric
In statistics, a distribution is considered symmetric if the data set that is mound-shaped.symmetric histogram
For a symmetric histogram, the values of the mean, median, and mode are all the same and are all located at the center of the distribution.undefined bimodal
A undefined bimodal histogram has a shape is not specifically defined, but we can note regardless that it is bimodal, having two separated classes or intervals equally representing the maximum frequency of the distribution.uniform
A uniform shaped histogram indicates data that is very consistent; the frequency of each class is very similar to that of the others.unimodal
If a data set has only 1 value that occurs most often, the set is called unimodal.Image Attributions
Description
Learning Objectives
Here you'll learn to understand and interpret frequency tables and histograms.