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7.03 Review: Measures of center and spread

Lesson

We have already learned about measures of center and spread .  Measures of central tendency attempt to summarize a set of data with a single value that describes the center or middle of the data values.  

The three main measures of central tendency are the meanmedian, and mode. Deciding which one is best depends on the characteristics of the particular set of data, as we already saw with the mean.

 

Measures of center

Mean

  • The numerical average of a data set, this is the sum of the data values divided by the number of data values.
  • Appropriate for sets of data where there are no values much higher or lower than those in the rest of the data set

 

Median

  • The middle value of a data set ranked in order
  •  A good choice when data sets have a couple of values much higher or lower than most of the others

 

Mode

  • The data value that occurs most frequently
  • A good descriptor to use when the set of data has some identical values, when data is non-numeric (categorical) or when data reflects the most popular item

 

Median

The median is one way of describing the middle or the center of a data set using a single value. The median is the middle value in a data set. 

Suppose we have five numbers in our data set: $4$4, $11$11, $15$15, $20$20 and $24$24.

The median would be $15$15 because it is the value right in the middle. There are two numbers on either side of it.

$4,11,\editable{15},20,24$4,11,15,20,24

 

If we have an even number of terms, we will need to find the average of the middle two terms. Suppose we wanted to find the median of the set $2,3,6,9$2,3,6,9, we want the value halfway between $3$3 and $6$6. The average of $3$3 and $6$6 is $\frac{3+6}{2}=\frac{9}{2}$3+62=92, or $4.5$4.5, so the median is $4.5$4.5.

$2,3,\editable{4.5},6,9$2,3,4.5,6,9

If we have a larger data set, however, we may not be able to see right away which term is in the middle. We can use the "cross out" method and cross out the numbers on the ends until we get to the middle value.


Practice questions

Question 1

Find the median of this set of scores:

$11$11, $11$11, $13$13, $14$14, $18$18, $22$22, $23$23, $25$25

Question 2

Given the following set of scores:

$65.2,64.3,71.6,63.2,45.2,62.2,46.8,58.7$65.2,64.3,71.6,63.2,45.2,62.2,46.8,58.7

  1. Write the list of scores in ascending order.

  2. Calculate the median.

 

Mode

The mode is another measure of central tendency - that is, it's a third way of describing a value that represents the center of the data set. The mode describes the most frequently occurring piece of data. 

Let's say we ask $10$10 people how many pets they have. $2$2 people say no pets, $6$6 people say one pet and $2$2 people say they have two pets. What is the most common number of pets for people to have? In this case, the most common number is $1$1 pet, because the largest number of people $\left(\frac{6}{10}\right)$(610) had one pet. So the mode of this data set is $1$1.

 

Practice questions

Question 3

Find the mode of the following scores:

$2,2,6,7,7,7,7,11,11,11,13,13,16,16$2,2,6,7,7,7,7,11,11,11,13,13,16,16

  1. Mode = $\editable{}$

Question 4

Find the mode of the following scores:

$8,18,5,2,2,10,8,5,14,14,8,8,10,18,14,5$8,18,5,2,2,10,8,5,14,14,8,8,10,18,14,5

  1. Mode = $\editable{}$

 

Range

The range is a measure of spread in a numerical data set. In other words, it describes whether the values in a data set are very similar and clustered together, or whether there is a lot of variation in the data values and they are very spread out.

If we looked at the range of ages of students in a $6$6th grade class, everyone would likely be between $11$11 and $13$13, so the range is $2$2 because we subtract the largest age from the smallest age ($13-11$1311). This is quite a small range.

However, if we looked at the ages of people waiting at a bus stop, the youngest person might be a $2$2 year old and the oldest person might be a $90$90 year old. The range in this set of data is $88$88 ($90-2$902) which is quite a large range.

To calculate the range

Subtract the least value in the set from the greatest value in the set.

 

Practice questions

Question 5

Find the range of the following set of scores:

$10,7,2,14,13,15,11,4$10,7,2,14,13,15,11,4

Question 6

The range of a set of scores is $8$8, and the greatest score is $19$19.

What is the least score in the set?

 

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