Averages – Mean, Median, Mode (With Examples)

Key Concepts

• Averages
• Median, mode and range
• Making and interpreting line plots
• Stem and leaf plots
• Outcomes
• Probability as a fraction

Introduction: 

In this chapter, we will be learning about the following:   

• Finding averages from the data. 
• Finding mean, median, mode and range of data. 
• Probability of a fraction. 
• Finding outcomes from the data. 

Averages 

Average: An average is defined as the sum of all the values divided by the total number of values in each set. It is also known as the arithmetic mean. 

5.1.1 Finding the averages from the data 

Example:  Find the average of 3, 5 and 7. 

Sol.: 

 Step-1: Find the sum of the numbers. 

3 + 5 + 7 = 15 

 Step-2: Calculate the total number. 

   There are 3 numbers. 

Step-3: Finding average. 

  = 3 

Example: Total height of a class is 1300 cm. If the average height of a class is 65 cm, find the number of students in the class. 

Sol.: Total height of a class = 1300 cm  

             Average = 65 cm 

No. of students =   

   = 20 

Median, mode and range 

Finding the median of a data set 

Median: The median is the middle value of a sorted list of numbers.  

To find the middle value, numbers must be listed in numerical order from the smallest to largest. So, you may have to sort your list before you can find the median. 

Median for ungrouped data: 

Step-1: Arrange the data in ascending or descending order. 

Step-2: Let the total number of observations be n.  

To find the median, we need to consider whether n is even or odd. 

=> If n is odd, then use the following formula: 

Example: Find the median of 14, 63 and 55. 

Sol.: Arranging in ascending order, we get 14, 55, 63. 

Here, no. of observations = 3 

So,  

Median = 2^nd observation 

Median = 55 

=> If n is even, then use the formula: 

Median =   

Example: Find the median of 50, 67, 24, 34, 78, 43. 

Sol.: Arranging in ascending order, we get 24, 34, 43, 50, 67, 78   . 

Here, n = 6 

                             = 3 

Using the median formula 

  =  

Median = 46.5 

Finding the mode of a data set 

Mode: The value which appears most often in the given data, i.e., the observation with the highest frequency, is called the mode of data. 

Ungrouped Data: For ungrouped data, we need to identify the observation which occurs maximum times. 

Example: Find the mode of 13, 18, 13, 14, 13, 16, 14, 21, 13. 

Sol.: The mode is the number that is repeated more often than another so, mode = 13 

Example: Find the mode of 14, 12, 13, 4, 5, 6, 5, 4, 5, 2, 5 

Sol.: The value 5 appears the greatest number of times.  

Thus, mode = 6.  

Finding the range of a data set 

Range: The range is the difference between the lowest and highest values. 

Range = greatest number – least number. 

Example: Find the range of 4, 5, 6, 7, 8, 9 

Sol.:       The lowest value = 4 

The highest value = 9 

                      Range = 9 – 4 = 5 

Example: Find the range of 100, 10, 20, 60, 90, 200. 

Sol.:       The lowest value = 10  

The highest value = 200 

         Range = 200 – 10 = 190 

Make and Interpret line plots. 

Line plot: A line plot can be defined as a graph that displays data as points or checkmarks above a number line, showing the frequency of each value. 

Example: The table shows the number of ribbons and length of ribbons (Inches). 

Draw a line plot. 

Sol.: 

       Length of ribbons (In)   

(a)  How many ribbons are there altogether?  

        Sol.: 3 + 5 + 2 + 1 = 11 

(b) Find the range of the length of the ribbon. 

        Sol.: Highest length = 10 

                  Lowest length = 6 

      Range = 10 – 6 = 4  

(c) Find the mean of the ribbons. 

       Sol.: 

Stem and leaf plots 

Organise and represent data in a stem and leaf plot 

Stem and leaf plots: A stem-leaf plot is a special table where each data value is split into a ‘stem’ (the first digit or digits) and a ‘leaf’ (usually the last digit). 

Example: Prepare stem and leaf plot for the given data. 

34, 40, 52, 57, 57, 60, 60, 63, 67, 69, 69, 69 

Sol.: 

Example: Make stem and leaf plots from the given data. 

      26, 37, 48, 33, 49, 26, 19, 26, 48 

Sol.: 

Using stem and leaf plot to find median, mode and range. 

Example: Find the median, mode, mean and range from the following data: 

Sol.: The data values are 20, 32, 32, 35 and 41. 

=> Mean of the data =   

   Mean = 32 

=> Median = the middle value of the data  

Median = 32 

=> Mode = the value that frequently occurs  

Mode = 32 

=> Range = highest value – lowest value 

= 41 – 20 

= 21 

Outlier: An outlier is any number in the data that is much far away from the largest group of data. 

The outlier in this set of data is 41. 

Outcomes 

Outcome: A possible result of a probability experiment. 

Types of outcomes: 

1) Impossible outcome: If a particular outcome can never result from an event, then it is an impossible outcome. 

Example: Suppose you have the bag with all red marbles. 

So, if you pick a marble from the bag without looking, what is the probability that you would pick a blue marble? 

Sol.: In this event, blue marble is an impossible outcome because there is no blue marble in the bag. So, the probability of picking a blue marble is zero. 

2) Unlikely or least likely outcome: 

The outcome of an event is least likely to occur when it has a lesser chance of happening than all other outcomes in the event. 

Example: Suppose you have the bag with the red, black and blue marbles. 

If you pick a marble from the bag without looking. What is the probability that you would pick a blue marble? 

Sol.: There are 3 – blue marbles 

 4 – red marbles 

 4 – black marbles 

So, when you pick a marble from the bag, you could get any of the 3 colour marbles. However, since there are fewer blue marbles than either of the other two-colour marbles in the bag, it is least likely that you will pick a blue marble. 

Hence, picking a blue marble is the least likely outcome. 

3) Equally likely outcome: The outcomes of an event is equally likely to occur when they have exactly an equal chance of happening in the event. 

Example: Suppose you have a bag with red and white marbles. 

If you pick a marble from the bag without looking, what is the probability that you would pick a white marble? 

Sol.: There are 4 red and white marbles in the bag. 

So, picking a white marble is as likely as picking a red marble. 

Both the outcomes of this event are equally likely because there is an equal number of red and white marbles in the bag. 

4) Most likely outcome: The outcome of an event is most likely to occur when it has a greater chance of happening than all other outcomes in the event. 

Example: Suppose a bag contains black and green marbles. 

If you pick a marble from the bag without looking, what is the probability that you would pick a green marble? 

Sol.: There are 8 – green marbles 

 3 – black marbles in the bag 

So, picking a green marble is the most likely outcome because there are more green marbles than black marbles in the bag. 

5) Certain outcome: The outcome of an event is certain to occur when it always happens in the event. 

Example: Suppose a bag contains all yellow marbles in it. 

If you pick a marble from the bag without looking, what is the probability that you would pick a yellow marble? 

Sol.: There are only yellow marbles in the bag, so picking a yellow marble is a certain outcome because all the marbles in the bag are yellow. 

Probability as a fraction 

Determining the probability of an event: 

Probability: A probability event can be defined as a set of outcomes of an experiment. 

Probability of an event: The number of favourable outcomes to the total number of outcomes is defined as the probability of occurrence of any event. So, the probability that an event will occur is given as 

Example: A bag contains 5 green pens, 3 blue pens, 8 black pens and 4 red pens. A pen is picked at random. What is the probability that the pen is green? 

Sol.: There are 5 + 3 + 8 + 4 = 20 pens in the box 

P(green)  

Expressing the probability as a fraction: 

Probability can be described by a number written as a fraction. 

Example: What is the probability of the spinner getting a gift box from the following wheel?     

Example: When you toss a number cube, what is the probability of rolling a 3? 

Sol.: P (3) =    

Real-world problems: Data and probability 

Example: A die is rolled. Find the probability that an even number is obtained. 

Sol.: Let us first write sample space. 

S = {1, 2, 3, 4, 5, 6} 

Let E be the event in which “an even number is obtained”.  

E = {2, 4, 6} 

P(E) =   

Example: Midterm exam scores for a class are 87, 99, 75, 87, 94, 75, 35, 88, 87, 93. 

Find the mean, median, mode and range. 

(a) Mean =    

 (b) Median = Arranging data is ascending order 

35, 75, 75, 87, 87, 87, 88, 93, 94, 99    

The median is 87. 

 (c) Mode = 87. It occurs 3 times. 

 (d) Range = higher value – lowest value 

                        = 99 – 35  

         Range = 64 

Exercise:

1. What is the median of the following sample

               5, 5, 11, 9, 8, 5, 8?

2. Find the mean of 42, 40, 40, 47, 41.
3. The following are the scores made on a math test 80, 90, 90, 85, 60, 70, 75, 85, 90, 60, 80.
   • What is the mode of these scores?
4. Find the range of the data 2, 7, 11, 12, 18, 23, 25, 35, 27, 33.
5. Sophia saw the line plot of coronavirus cases varying in the first 10 days

Find the day when the number of cases reported is 3 thousand.

6. Make stem and leaf plots for the given data

74, 88, 97, 72, 79, 86, 95, 79, 83, 91.

7. A spinner has 4 equal sectors coloured yellow, blue, green and red. What is the probability of  landing on purple after spinning the spinner?

8. A single 6 sided die is rolled. What is the probability of rolling a number less than 7?
9. The bag contains gumballs.
   1. If you pick a gumball from the bag without looking, what is the probability that you would pick an orange gumball?
   2. If you pick a gumball from the bag without looking, what is the probability that you would pick a green gumball?

10. Find the probability of tossing a coin.   
11. Write three out of five in fraction form.
12. Complete the mean of the first 6 odd, natural numbers.
13. Look at the table below
    1. Which kind of bead is the least likely to be picked?
    2. Which kind of bead is the most likely to be picked?

Kinds of beads in a grab bag

Metal	Glass	Clay	Wood	Plastic
10	12	3	4	1

14. Two coins are tossed. Find the probability that two heads are obtained.

What we have learnt:

• How to find the averages from the given data
• How to find the median, mode and range of a data set
• Make and interpret line plots.
• Organize and represent data in a stem and leaf plot.
• Using stem and leaf plot find median, mode and range.
• What probability is by predicting the outcome of planned experiments.
• To describe events as likely or unlikely and discussed the degree of likelihood using words such as certain, equally likely, and impossible.
• To understand the mean, median, mode, and range.
• How to express outcome as a fraction.
• How to solve problems based on probability.

Concept Map