Category: Statistics

What is the Mean in a Data Set?

You will see at least one question on your entrance exam that asks: What is the mean?

To find the mean in a data set, you must add up all of the values in the set, and then divide by the amount of items in the set.

The mean is sometimes referred to as the arithmetic mean or the average.

Exam questions asking you to find the mean will provide the data in narrative form or in charts or tables.

For example, consider the following data set: 10, 11, 14, 12, 15

The mean for this data set is calculated as shown below.

STEP 1: Find the total of the values.

10 + 11 + 14 + 12 + 15 = 62

STEP 2: Divide the total from step 1 by the amount of items in the set to find the mean.

Here we have 5 numbers in the set.

62 ÷ 5 = 12.4

So, the mean is 12.4.

The mean should not be confused with the median or the mode.

What is the Mean If the Data Set Changes?

If one or more values is removed from or added to your data set, you will need to recalculate the mean.

If you add a value greater than the mean, the mean will increase.

On the other hand, the mean will decrease if you add a value that is less than the existing mean.

The mean will increase if you remove a value from the data set that is less than the mean.

However, the mean will decrease if you remove a value that is greater than the mean.

What is the Mean If Items Are Weighted?

You may see problems that state that certain items in a mean calculation carry more weight than others.

For example, your grade for a class may be determined by taking the average of your project scores by 25% and the average of your test scores by 75%.

So, if you received an average score of 70 on your projects and an average score of 80 on your exams, your grade is calculated as follows:

(70 × .25) + (80 × .75) = 17.50 +  60 = 77.50

Exam questions may also ask you how to correct an erroneous mean calculation.

You may also want to try our other free statistics exercises.

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Median – Definition and Explanation

The median is the middle value in a data set when the numbers are organized from smallest to largest.

Since the median is in the middle, it divides the data set into two groups.

The first group is the 50% of the data, excluding the median, that is less than the middle value.

The second group is the 50% of the data, excluding the median, that is greater than or equal to the middle value.

Exam questions on the median may simply give a list of values.

Other exam problems will provide practical problems with data sets.

How to Find the Median – Odd Number of Items

The median is easy to find when a data set has an odd number of items, such as 13 or 15 items.

When the group consists of an odd number of items, one item will be exactly in the middle.

Look at the example data set below.

15, 2, 14, 5, 9, 7, 6, 11, 12, 4, 10, 14, 8

We can see that there are 13 items in the above data set.

STEP 1:

To learn how to find the median on exam problem like this, you should first put the numbers in the group in ascending order.

So, first of all, reorder the numbers from lowest to highest.

2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 14, 15

STEP 2:

Since this set has 13 items, the median value is the 7th item in the reordered list.

In other words, six numbers are less than the middle value and six numbers are greater than the middle value.

2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 14, 15

So, the median is 9.

How to Find the Median – Even Number of Items

When the data set consists of an even number of items, like 10 or 12 items, the median is a little more difficult to find.

That is because in this case, there will be two items in the middle.

Look at the example data set that follows.

29, 35, 24, 31, 36, 39, 28, 22, 33, 37, 36, 25, 26, 30

We can see that the data set above contains 14 numbers.

STEP 1:

Remember to put the data set in ascending order first of all.

22, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 36, 37, 39

STEP 2:

When the group has an even number of items, you will need to find the two values that are in the middle of the set.

Our data set in this problem has 14 items, so we need the 7th and 8th items in the set.

22, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 36, 37, 39

We can check this by counting that there are 6 items in the list that are less than 30:

22, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 36, 37, 39

There are also 6 items in the list that are greater than 31:

22, 24, 25, 26, 28, 29, 30, 31, 33, 35, 36, 36, 37, 39

STEP 3:

Add the two middle values together and then divide this result by 2.

30 + 31 = 61

61 ÷ 2 = 30.5

So, the median is 30.5.

You may also want to see our posts on range, mode, and mean.

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What is Mode in Statistics?

You will need to be able to answer the question “What is Mode” on your entrance exam. The mode is the most frequently occurring value within a data set. The basic definition of mode is that it is the most common value in a set of numbers.

[WpProQuiz 41]

What is Mode – Example:

For example, look at the data set: 8, 2, 3, 6, 7, 9, 4, 8, 5, 9, 8, 10, 6, 7

Here, we have a data set that contains 14 numbers.

To find the mode, we need to determine which of the values occurs the most frequently in the set.

For data sets with 20 or fewer items, it can be helpful to put the numbers in ascending order before trying to find the mode.

Our data set is: 8, 2, 3, 6, 7, 9, 4, 8, 5, 9, 8, 10, 6, 7

In ascending order, it looks like this: 2, 3, 4, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10

When the numbers are in order from lowest to highest, we can easily see that the number 8 occurs three times.

Other numbers occur only one or two times in the list.

So, the mode is 8.

For larger data sets, you simply have to try to count which of the numbers occurs the most frequently.

You may need to draw a line through the numbers you have counted or keep a tally in order to keep track.

You can also set up a frequency table in order to determine the mode.

Multiple Modes and No Mode

So, what happens if two or more values occur the same number of times? Is there a mode?

Consider the following data set: 1, 3, 3, 4, 4, 5, 6

The modes of this data set are 3 and 4.

In other words, we can say that the data set is bimodal.

So, a data set can have multiple modes.

It is also possible for a data set to have no mode, if no value appears more than once in the data set.

What is Mode – Practical Problems

Questions about mode on your entrance exam are usually worded as practical problems.

You will see questions including data sets on weights, measurements, temperatures, ages, scores, incomes, population sizes, and other numerical values.

The data may be given in narrative form in your exam question.

Alternatively, you might see data summarized in a chart or table.

You exam question might ask: What is the mode?

Exam questions might also be worded as follows: What is the modal value?

You can learn more about mode from the Math is Fun website.

What is Range in Math? – Definition

Range in math is defined as a measurement of the spread of data.

“Spread” relates to how scattered the data looks when plotted on a graph.

Range is a very simplistic measurement of data since it compares only the largest and smallest values.

Determining the Range

The range is the difference between the highest value and the lowest value in a data set.

In order to calculate the range, we simply subtract the lowest value from the highest value.

Range = Highest Value – Lowest Value

To put it another way, the range is calculated by subtracting the minimum value in the data set from the maximum value in the data set.

For example, if the lowest number is a data set is 51 and the highest value is 87, the range is 36 since 87 – 51 = 36.

Types of Exam Questions on Range in Math

You will see different types of questions on range on your entrance exam.

You may see questions that simply provide a set of numbers and then ask you to determine the range.

The numbers in the set will not usually be listed in ascending order like this:  3, 4, 6, 7, 8, 9, 12

In other words, the numbers in the data set will normally be mixed up like this:  8, 3, 9, 4, 12, 7, 6

For problems on range in math, the values may be provided in a table or chart.

However, the most common types of questions on range in math are practical problems in narrative form.

Other Measurements of Data

Entrance exams also have questions other measurements of data.

These measurements include mean, mode, median, and quartiles.

Mean is the average of the values.

Mode is the value that occurs most frequently.

Median is the value that is in the middle of the data set.

Quartiles are formed when we place the values of the data set into four groups.

We will look at mean, mode, median, and quartiles in separate posts.

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What is a Frequency Table?

The definition of the word “frequency” is how often something occurs.

So, a frequency table shows how many occurrences have taken place.

You may need to count items or events to determine their frequency.

The counted items are also called data.

The data is placed into groups or categories.

Frequency tables usually have two or more columns.

The first column shows the category and the second column shows the frequency.

You may also see three column frequency tables.

In three column tables, the first column shows the category, the second column shows the tally or count, and the third column shows the frequency.

Example of a Frequency Table

Suppose that you have to keep track of the amount of rain in your town each day for two weeks.

At the end of the two weeks, you have recorded the following amounts in inches per day:

• Day 1:    2.1
• Day 2:    1.0
• Day 3:    0.5
• Day 4:    0.0
• Day 5:    0.2
• Day 6:    0.1
• Day 7:    1.4
• Day 8:    2.0
• Day 9:    2.2
• Day 10:  1.1
• Day 11:  0.8
• Day 12:  0.7
• Day 13:  0.6
• Day 14:  1.8

To make a frequency table, you have to sort your results into groups.

So, this type of table is called a grouped frequency table.

Rainfall in My Town for 2 Weeks
Amount in inches		Tally	Frequency
0 to 0.5		||||	4
0.6 to 1.0		||||	4
1.1 to 1.5		||	2
1.6 to 2.0		||	2
2.1 to 2.5		||	2

In other words, there were 4 days when the amount of rainfall was between 0 and 0.5 inches.

There were 4 days when the amount of rainfall was between 0.6 and 1.0 inches, and so on.

Frequency Table – Exercise

Now look at the following data and then complete the frequency table.

The answer is provided after the exercise.

Exercise: A basketball team records the number of points it scores in each game of the season. Complete the frequency table for the following data.

Basketball Games This Season
Points		Tally	Frequency
0 to 25
26 to 50
51 to 75
More than 75

• Game 1:    25
• Game 2:    36
• Game 3:    55
• Game 4:    24
• Game 5:    70
• Game 6:    48
• Game 7:    59
• Game 8:    76
• Game 9:    32
• Game 10:  41

Answer to Frequency Table Exercise

Remember to count up the items for each group in the “tally” column, and then write the number in the “frequency” column.

Basketball Games This Season
Points		Tally	Frequency
0 to 25		||	2
26 to 50		||||	4
51 to 75		|||	3
More than 75		|	1

Frequency Tables for Continuous Data

In many cases, you will need to think carefully when setting up categories for your data.

You need to be sure that your categories will include the results for all of your data.

Consider the following example.

Example:  Eight students take part in a race. Their times in seconds are shown below.

• Student 1:   41.25
• Student 2:   36.29
• Student 3:   29.14
• Student 4:   41.38
• Student 5:   35.92
• Student 6:   20.99
• Student 7:   37.02
• Student 8:   38.64

Summary of Race Results in Seconds
Seconds		Tally	Frequency
0 to 20
21 to 35
36 to 40
41 to 65

If we try to use the above table for our results, we will not be able to place the results for student 5 and student 6 into any of the above categories.

That is because student 5’s result of 35.92 seconds does not fit into either the “21 to 35 seconds” category or the “36 to 40 seconds” category.

In addition, student 6’s result of 20.99 seconds does not fit into either the “0 to 20 seconds” category or the “21 to 35 seconds” category.

Using Inequalities in Frequency Tables

We can solve this problem by using inequalities to ensure that our categories are continuous.

Exercise: Fill in the frequency table below. Variable “S” represents the result in seconds. The students’ results are provided again for ease of reference.

Summary of Race Results in Seconds
Seconds		Tally	Frequency
0 ≤ S < 20
20 ≤ S < 35
35 ≤ S < 40
40 ≤ S < 65

 

• Student 1:   41.25
• Student 2:   36.29
• Student 3:   29.14
• Student 4:   41.38
• Student 5:   35.92
• Student 6:   20.99
• Student 7:   37.02
• Student 8:   38.64

When you have completed the table, check your answer below.

Answer to Continuous Frequency Table Exercise

Summary of Race Results in Seconds
Seconds		Tally	Frequency
0 ≤ S < 20			0
20 ≤ S < 35		||	2
35 ≤ S < 40		||||	4
40 ≤ S < 65		||	2

Notice how student 5’s result of 35.92 seconds now fits into the “35 ≤ S < 40 seconds” category.

Student 6’s result of 20.99 seconds fits into the “20 ≤ S < 35 seconds” category.

When you have completed the exercises on this page, you may also want to look at our separate exercise on inequalities.

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