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## Range, variance, standard deviation

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Understanding the Mean

Have you ever been on a track team? Take a look at this dilemma.

“It is time to get ready for competition!” Mr. Watson the track coach said to his track and field team on Monday afternoon.

“What does that mean coach?” Marco asked smiling a huge grin.

The Hawks were very excited that their season had gone so well and now they were ready to prepare for regionals.

“It means that we are all going to figure out where we are in our team standings and then set goals to improve. That way we can have a great showing at regionals,” Mr. Watson explained.

As soon as he heard this, Alfredo began to figure out his standing. Alfredo is a high-jumper at his school. He has 8 teammates whose records are 172 cm, 174 cm, 175 cm, 179 cm, 181 cm, 181 cm, 182 cm, and 185 cm. If Alfredo’s record is 176 cm, how does he compare to the rest of the team?

You can help Alfredo figure this out by understanding statistics and data. Pay close attention during this Concept and you will see how to solve this dilemma.

### Guidance

In the real world, there are many situations in which a large group of data is collected. In order to make sense of the data, we use a number of statistical measures . These measures help us to generalize a group of data, make inferences about it, and compare it with other groups of data. These statistical measures include mean, median, mode, range, deviation from the mean, and absolute deviation from the mean. Depending on the situation, certain measures may be more helpful than others in interpreting data.

Let’s look at these statistical measures.

The mean , median , and mode are three common measures of central tendency ; they are three mathematical tools frequently used to analyze data. The mean , commonly referred to as the average, is the sum of all the data items divided by the number of data items. The median is the middle number in a set of data that is ordered from lowest to highest. If there is an even number of data, we take the average of the middle two numbers to find the median. Finally, the mode is the number that occurs most often.

What happens if we want to find a difference between the mean and another value? Take a look at this dilemma.

At a restaurant, the food servers report how much money in tips they earn each night. One Saturday, the food servers reported the following tips: $45,$37.50, $51,$89, $47, and$55. Greg is the newest food server and he reported $51. He wants to know how well he did compared with the others. Do you know how to figure this out? One way to do this is to find the deviation from the mean . This tells you how far away from the mean, or average, each food server was. Here are the steps. First, find the mean. Then find the difference between each number and the mean by subtracting. Because we want to find “how far away” each person was, it is like finding a distance—we only use positive numbers. Step 1: Find the mean. $& = \frac{45+37.5+51+89+47+55}{6}\\&= \frac{324.5}{6}\\&= 54.08$ The average food server tip was$54.08.

Step 2: Find the difference of each food server tip from the mean. This is the deviation from the mean. Notice that Greg’s value is always used because we are looking for the deviation between the tips of the other waiters and Greg.

Difference from mean Deviation from the mean
54.08 - 37.5 16.58
54.08 - 45 9.08
54.08 - 47 7.08
54.08 - 51 3.08
55 - 54.08 .92
89 - 54.08 34.92

When we subtract, we place the largest of the two numbers first so that the difference is positive.

The deviation from the mean can let each food server know how far he or she was from the average tips that night.

Greg earned $51. The average was$54.08 so he was very close to the average although a little bit lower.

Write these steps down in your notebook.

Remember that when we look for the deviation from the mean, we are looking for the difference between an average and a value.

Do you know what the mean absolute deviation is?

The range is found by subtracting the smallest number from the largest number. This will give you an idea of the span or the breadth of the data.

For instance, if you are a new car buyer and have just entered the work force, it may help for you to know that the mean price of new cars is $22,300. This may be too high for you, but the mean doesn’t really supply you with enough information. If you know that new car prices vary from$10,500 to \$89,900, then you have a better idea about what a car might cost at the lower end, easier for your income level. The range of car prices, in this example is $\89,900 - \10,500 = \79,400$ . It is a broad range that should allow for you to purchase a car.

A second statistical measure that can be useful is the mean absolute deviation . We already found the deviation from the mean, which is how far an individual item is from the mean. Because we only found the positive difference, we actually found the absolute deviation . The mean absolute deviation , then, is the mean of those deviations .

I know that it seems confusing, try to think of it in another way.

Let’s suppose that a school teacher gives an exam. The mean in the class is 82%. That seems pretty good, right? Doesn’t that mean that the average student got a B? Not necessarily. If the mean absolute deviation is high, it means that the average student either did much better or much worse than 82%. It indicates a greater range of scores. On the other hand, if the mean absolute deviation is low, it is a stronger indication that the group as a whole is working around the B level.

A city surveyor took elevation measurements around a coastal city that has a reported mean elevation of 35 feet above sea level. He went to various homes and gathered the following data: 152, 316, 26, 64, 20, 506, 210, and 89. Find the range and mean absolute deviation.

Step 1: Find the range. Subtract the smallest number from the largest.

$506 - 20 = 486$

Step 2: Find the deviations from the mean (mean of 35 already given).

Difference from mean Deviation from the mean
35 – 20 15
35 – 26 9
64 – 35 29
89 – 35 54
152 – 35 117
210 – 35 175
316 – 35 281
506 – 35 471

Step 3: Find the mean of the deviations from the mean.

$& = \frac{15+9+29+54+117+175+281+471}{8}\\&= \frac{1151}{8}\\&= 143.9$

The mean absolute deviation is about 143.9 feet.

Use the following information to answer each question.

The following scores were earned on a math test: 65, 70, 82, 83, 50, 90 and 88. Kara earned a 82.

#### Example A

What is the mean of the scores?

Solution: $75$

#### Example B

Find the deviation from the mean for each score.

Solution: $10, 5, 7, 8, 25, 15, 13$

#### Example C

How did Kara's score compare?

Solution: Kara's score was above the mean by 7 points.

Now let's go back to the dilemma from the beginning of the Concept.

In this case, finding the deviation from the mean would be most useful in answering the question because he wants to compare his individual record to that of the team.

Step 1: Find the mean.

$& = \frac{172+174+175+179+181+181+182+185}{8}\\&= \frac{1429}{8}\\&= 178.6$

Step 2: Calculate his deviation from the mean: $178.6 - 176 = 2.6cm$ .

Alfredo’s record is 2.6cm lower than the mean. He should continue to work on improving his high jump. Now that Alfredo understands his place on the team, he can work with the coach on a plan to improve.

Notice that when comparing to others, we use the deviation from the mean.

### Vocabulary

Statistical Measures
measures used to generalize a set of data, make inferences and compare it with other groups of data.
Measures of Central Tendency
Math tools used to analyze data
Mean
the average of a set of data.
Median
the middle score in a set of data that has been arranged from smallest to largest.
Mode
the value that occurs the most times in a data set.
Deviation from the mean
how far a value is from the mean or average
Range
the breadth of the data, the difference between the largest and smallest values.
Mean Absolute Deviation
the mean of the deviations

### Guided Practice

Here is one for you to try on your own.

The town keeps statistics on its local races. Here are the times from a recent 5k race. The average time was 23 minutes.

${21, 21, 22, 18, 19, 23, 25, 27, 30}$

What is the mean absolute deviation?

Solution

To figure this out, we must first find the range.

$30 - 21 = 9$

Next, we find each deviation of the mean. We do this by subtracting each time from the mean of 23 minutes.

$2, 2, 1, 5, 4, 0, 2, 4, 7$

Now, we find the mean of the deviations.

$3$

The mean absolute deviation of the times is $3$ minutes.

### Practice

Directions: Define each term.

1. Mean
2. Median
3. Mode
4. Deviation from the mean
5. Range

Directions: Use this data to complete the following questions.

Two groups of adult female harbor seals were weighed from different parts of the globe, one from the Pacific Ocean and one from the Atlantic Ocean.

The Pacific Ocean group had weights of: 126kg, 130kg, 135kg, 136kg, 137kg, 140kg, 148kg, and 150kg. The Atlantic Ocean group had weights of 117kg, 119kg, 122kg, 123kg, 130kg, 131kg, 141kg, 149kg, and 152kg. A marine biologist decided to gather data to compare the two groups.

Beginning with the Pacific Ocean Group.

1. What is the mean of the data set?
2. What is the median?
3. What is the mode?
4. What is the range?
5. If a new seal was weighed with a weight of 137kgs, what would be the deviation from the mean?

Now with the Atlantic Ocean Group.

1. What is the mean of the data set?
2. What is the median?
3. What is the mode?
4. What is the range?
5. If a new seal was weighed with a weight of 137kgs, what would be the deviation from the mean?

### Vocabulary Language: English

Chebyshev's theorem

Chebyshev's theorem

Chebyshev’s Theorem gives us information about the minimum percentage of data that falls within a certain number of standard deviations of the mean, and it applies to any population or sample, regardless of how that data set is distributed.
descriptive statistics

descriptive statistics

In descriptive statistics, the goal is to describe the data that found in a sample or given in a problem.
deviation

deviation

Deviation is a measure of the difference between a given value and the mean.
Dispersion

Dispersion

The dispersion is equal to the range of a given set of data.
Interquartile range

Interquartile range

The interquartile range is the difference between the third quartile and the first quartile (Q3-Q1).
Mean

Mean

The mean of a data set is the average of the data set. The mean is found by calculating the sum of the values in the data set and then dividing by the number of values in the data set.
measures of central tendency

measures of central tendency

The mean, median, and mode are known as the measures of central tendency.
Median

Median

The median of a data set is the middle value of an organized data set.
Mode

Mode

The mode of a data set is the value or values with greatest frequency in the data set.
Range

Range

The range of a data set is the difference between the smallest value and the greatest value in the data set.
Sampling error (random variation)

Sampling error (random variation)

Sampling error occurs whenever a sample is used instead of the entire population, where we have to accept that our results are merely estimates, and therefore, have some chance of being incorrect.
standard deviation

standard deviation

The square root of the variance is the standard deviation. Standard deviation is one way to measure the spread of a set of data.
Statistics

Statistics

Statistics is a branch of mathematics that involves collecting, analyzing and displaying data.
variance

variance

A measure of the spread of the data set equal to the mean of the squared variations of each data value from the mean of the data set.