AveragesInstatistics, an average is defined as the number that measures the central tendency of a given set of numbers. There are a number of different averages including but not limited to: mean, median, mode and range.MeanMean is what most people commonly refer to as an average. The mean refers to the number you obtain when you sum up a given set of numbers and then divide this sum by the total number in the set. Mean is also referred to more correctly as arithmetic mean.

Given a set ofnelements from a1to an

The mean is found by adding up all thea's and then dividing by the total number,n

This can be generalized by the formula below:

Mean Example ProblemsExample 1Find the mean of the set of numbers below

SolutionThe first step is to count how many numbers there are in the set, which we shall calln

The next step is to add upallthe numbers in the set

The last step is to find the actual mean by dividing the sum by n

Mean can also be found for grouped data, but before we see an example on that, let us first define frequency.Frequency in statistics means the same as in everyday use of the word. The frequency an element in a set refers to how many of that element there are in the set. The frequency can be from 0 to as many as possible. If you're told that the frequency an elementais 3, that means that there are 3as in the set.Example 2Find the mean of the set of ages in the table belowAge (years)Frequency

100

118

123

132

147

SolutionThe first step is to find the total number of ages, which we shall calln. Since it will be tedious to count all the ages, we can findnby adding up the frequencies:

Next we need to find the sum of all the ages. We can do this in two ways: we can add up each individual age, which will be a long and tedious process; or we can use the frequency to make things faster.Since we know that the frequency represents how many of that particular age there are, we can just multiply each age by its frequency, and then add up all these products.

The last step is to find the mean by dividing the sum byn

Population Mean vs Sample MeanIn theIntroduction to Statisticssection, we defined a population and asamplewhereby a sample is a part of a population.In statistics there are two kinds of means: population mean and sample mean. A population mean is the true mean of the entire population of the data set while a sample mean is the mean of a small sample of the population. These different means appear frequently in both statistics and probability and should not be confused with each other.Population mean is represented by the Greek letter (pronouncedmu) while sample mean is represented byx(pronouncedx bar). The total number of elements in a population is represented byNwhile the number of elements in a sample is represented byn. This leads to an adjustment in the formula we gave above for calculating the mean.

The sample mean is commonly used to estimate the population mean when the population mean is unknown. This is because they have the same expected value.MedianThe median is defined as the number in the middle of a given set of numbers arranged in order of increasing magnitude. When given a set of numbers, the median is the number positioned in the exact middle of the list when you arrange the numbers from the lowest to the highest. The median is also a measure of average. In higher level statistics, median is used as a measure of dispersion. The median is important because it describes the behavior of the entire set of numbers.Example 3Find the median in the set of numbers given below

SolutionFrom the definition of median, we should be able to tell that the first step is to rearrange the given set of numbers in order of increasing magnitude, i.e. from the lowest to the highest

Then we inspect the set to find that number which lies in the exact middle.

Lets try another example to emphasize something interesting that often occurs when solving for the median.Example 4Find the median of the given data

SolutionAs in the previous example, we start off by rearranging the data in order from the smallest to the largest.

Next we inspect the data to find the number that lies in the exact middle.

We can see from the above that we end up with two numbers (4and5) in the middle. We can solve for the median by finding the mean of these two numbers as follows:

ModeThe mode is defined as the element that appears most frequently in a given set of elements. Using the definition of frequency given above, mode can also be defined as the element with the largest frequency in a given data set.For a given data set, there can be more than one mode. As long as those elements all have the same frequency and that frequency is the highest, they are all the modal elements of the data set.Example 5Find the Mode of the following data set.

SolutionMode = 3 and 15Mode for Grouped DataAs we saw in the section on data, grouped data is divided into classes. We have defined mode as the element which has the highest frequency in a given data set. In grouped data, we can find two kinds of mode: the Modal Class, or class with the highest frequency and the mode itself, which we calculate from the modal class using the formula below.

where Lis the lower class limit of the modal class f1is the frequency of the modal class f0is the frequency of the class before the modal class in the frequency table f2is the frequency of the class after the modal class in the frequency table his the class interval of the modal classExample 6Find the modal class and the actual mode of the data set belowNumberFrequency

1 - 37

4 - 66

7 - 94

10 - 122

13 - 152

16 - 188

19 - 211

22 - 242

25 - 273

28 - 302

SolutionModal class = 10 - 12

where L= 10 f1= 9 f0= 4 f2= 2 h= 3therefore,

Solving the above using theorder of operations:

RangeThe range is defined as the difference between the highest and lowest number in a given data set.

Example 7Find the range of the data set below

Solution

Assumed MeanIn the section onaverages, we learned how to calculate the mean for a given set of data. The data we looked at was ungrouped data and the total number of elements in the data set was not that large. That method is not always a realistic approach especially if you're dealing with grouped data.That's where the assumed mean comes into play.Assumed mean, like the name suggests, is a guess or an assumption of the mean. Assumed mean is most commonly denoted by the lettera. It doesn't need to be correct or even close to the actual mean and choice of the assumed mean is at your discretion except for where the question explicitly asks you to use a certain assumed mean value.Assumed mean is used to calculate the actual mean as well as the variance and standard deviation as we'll see later.Assumed mean can be calculated from the following formula:

It's very important to remember that the above formula only applies to grouped data with equal class intervals.Now let us define each term used in the formula: x is the mean which we're trying to find. ais the assumed mean. his the class interval which we looked at in the section on data. fiis the frequency of each class, we find the total frequency of all the classes in the data set (fi) by adding up all thefi's Eachuiis found from the following formula:

wherehis the class interval and eachdiis the difference between the mid element in a class and the assumed mean.dis calculated from the following formula:

wherexis the midpoint of a given class.xis obtained from the following:

xiis the number in the middle of a given class.Thereforeuibecomes

Let's try an example to see how to apply the assumed mean method for finding mean.Example 1The student body of a certain school were polled to find out what their hobbies were. The number of hobbies each student had was then recorded and the data obtained was grouped into classes shown in the table below. Using an assumed mean of 17, find the mean for the number of hobbies of the students in the school.Number of hobbiesFrequency

0 - 445

5 - 958

10 - 1427

15 - 1930

20 - 2419

25 - 2911

30 - 348

35 - 402

SolutionWe have been given the assumed meanaas17and we know the formula for finding mean from the assumed mean as

we can find the class interval by using the class limits as follows:

We now have one component we need and we're one step closer to finding the mean.So we can solve the rest of this problem using a table where by we find each remaining component of the formula and then substitute at the end:HobbiesFrequency fixidi= xi- aui=dihfiui

0 - 4452-15-3-135

5 - 9587-10-2-116

10 - 142712-5-1-27

15 - 193017000

20 - 2419225119

25 - 29112710222

30 - 3483215324

35 - 402372048

fi= 200 fiui= -202

substituting

The mean number of hobbies is11.95.Tutorial on how to read and interpret pie charts.Example 1:

The pie chart below shows the percentages of blood types for a group of 200 people.

a) How many people, in this group, have blood type AB?

b) How many people, in this group, do not have blood type O?

c) How many people, in this group, have blood types A or B?

Solution to Example 1:a) 19% * 200 = 19 * 200 / 100 = 38 people

b) (100% - 40%) * 200 = 60 * 200 / 100 = 120 people

c) (16% + 25%) * 200 = 41 * 200 / 100 = 82

Example 2:

The pie chart below shows the percentages of types of transportation used by 800 students to come to school.

a) How many students, in the school, come to school by bicycle?

b) How many students do not walk to school?

c) How many students come to school by bus or in a car?

Solution to Example 2:a) 45% * 800 = 360 students

b) (100% - 15%) * 800 = 680 students

c) (30% + 10%) * 800 = 320 students

Example 3:

The pie chart below shows the percentages of the world population in each continent. The present world population is about 7 billion.

a) How many people live in Africa?

b) How many people do not live in Asia than?

c) How many more people live in North America than in South America?

Solution to Example 3:a) 15% * 7 = 1.05 billion

b) 7 - 60.4% * 7 = 2.772 billion

c) (7.3% - 6%) * 7 = 91 million

Example 4:

The total area of Asia, Africa, North America, South America, Europe and Australia is 134 million square kilometers. The pie chart below shows the percentages of each continent.

a) What is the area of Asia?

b) What is the area Europe?

c) How much bigger is Africa than Europe?

Solution to Example 4:a) 33.2% * 134 = 44.5 million square kilometers

b) 7.5% * 134 = 10.5 million square kilometers

c) (22.3% - 7.5%) * 134 = 19.8 million square kilometers

The bar graph given below shows the sales of books (in thousand number) from six branches of a publishing company during two consecutive years 2000 and 2001.Sales of Books (in thousand numbers) from Six Branches - B1, B2, B3, B4, B5 and B6 of a publishing Company in 2000 and 2001.

1.What is the ratio of the total sales of branch B2 for both years to the total sales of branch B4 for both years?

A.2:3B.3:5

C.4:5D.7:9

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2.Total sales of branch B6 for both the years is what percent of the total sales of branches B3 for both the years?

A.68.54%B.71.11%

C.73.17%D.75.55%

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3.What percent of the average sales of branches B1, B2 and B3 in 2001 is the average sales of branches B1, B3 and B6 in 2000?

A.75%B.77.5%

C.82.5%D.87.5%

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4.What is the average sales of all the branches (in thousand numbers) for the year 2000?

A.73B.80

C.83D.88

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5.Total sales of branches B1, B3 and B5 together for both the years (in thousand numbers) is?

A.250B.310

C.435D.560