Measures of Central Tendency, Dispersion, IQR and Standard Deviation

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Measures of Central Tendency, Dispersion, IQR and Standard Deviation. How do we describe data using statistical measures?. M2 Unit 4: Day 1. Statistics: numerical values used to summarize and compare sets of data.

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### Measures of Central Tendency, Dispersion, IQR and Standard Deviation

How do we describe data using statistical measures?

M2 Unit 4: Day 1

Measure of central tendency: a number used to represent the center or middle of a set of data values.

We will use 3 types:

• Mean
• Median
• Mode
Also called the average

given n numbers, it is the sum of the n numbers divided by n

is the sample mean

is the population mean

Mean:
Example
• Find the mean of the set of data.

75, 100, 63, 95, 82, 78

Median:

given n numbers, it’s the middle number when written in ascending order (from least to greatest)

n can be odd or even

Example
• Find the median of the data set.
• 1. Case 1(n is odd)

100, 95, 50, 78, 63, 75, 82

50, 63, 75, 78, 82, 95, 100

78 is the median

Example
• Find the median of the data set.
• 2. Case 2 (n is even)

82, 63, 100, 75, 78, 95

63, 75, 78, 82, 95, 100

You try: Find the median

3. 12, 13, 12, 10, 11, 14, 10

4. 1, 2, 4, 2, 5, 6, 1, 3

Mode

the number that occurs most frequently in a given data set

There are 3 cases:

1 Mode

No Mode

More than one Mode

Example: Find the mode

1. Find the mode of the data set: 1, 2, 3, 4, 5, 3

3 is the mode

2. Find the mode of the data set: 1, 2, 3, 4, 5

There is no mode

3. Find the mode of the data set: 1, 2, 3, 4, 5, 2, 3

2 and 3 are the modes

Measure of dispersion:

is a statistic that tells you how dispersed (spread out) the data values are.

One example of a measure of dispersion is range.

Range is difference between the largest and smallest data values

Example:
• Find the range of the data set:
• 63, 75, 78, 82, 95, 100

100 - 63 = 37

• Find the range of the data set:
• 21, 20, 26, 30, 16, 20

30 – 16 = 14

Interquartile Range (IQR)

The distance between the first and third quartiles

To calculate, find the median of the upper and lower half, then take the difference

Example

1. Find the IQR: 100, 78, 63, 50, 82, 95, 75

50, 63, 75, 78, 82, 95, 100

Find the median

Find the 1st and 3rd quartiles

IQR = 95 - 63

= 32

You Try:

Find the IQR: 78, 83, 91, 81, 111, 83, 72

72, 78, 81, 83, 83, 91, 111

Find the median

Find the 1st and 3rd quartiles

IQR = 91 - 78

= 13

Example

Find the IQR: 100, 63, 75, 82, 95, 100, 50, 78

50, 63, 75, 78, 82, 95, 100, 100

You Try:

Find the IQR: 2, 3, 2, 4, 1, 8, 5, 6

1, 2, 2, 3, 4, 5, 6, 8

Population Standard Deviation( “sigma”):
• measures the spread by looking at how far the observations are from their mean.
• The smaller the standard deviation, the less the data varies about the mean .
• The larger the standard deviation, the more the data will vary about the mean.
• is the population standard deviation
• is the sample standard deviation
Example:

Find the standard deviation for the following data set: 2, 5, 7, 11, 15

Find the standard deviation of the sample: 3, 4, 8, 9, 10

Example

A sample of 6 temperatures of patients was taken from all of the patients on wing E of the hospital.

The temperatures are:

98.6, 101, 97.8, 98, 99.4, 100.1

What is the standard deviation?

Example

A teacher looked at the GPAs of her advisement group.

The GPAs are:

95.3, 91.2, 86, 90.2, 82.2, 70.1, 72.3, 68.1, 75

Is the a sample or a population?

What is the standard deviation?

Example

A sample of 8 prices was taken from the menu of a given restaurant.

The prices are:

\$3.25, \$10.75, \$0.75, \$2.00, \$1.50, \$8.45, \$6.00, \$4.45

Is the a sample or a population?

What is the standard deviation?

Example

The following prices are for entrance into different sporting events at a given school.

The prices are:

\$4.00, \$2.50, \$3.00, \$5.00, 7.50

Is the a sample or a population?

What is the standard deviation?

Example :Compare the mean and standard deviation for the number of cars sold by the 2 dealers
• Dealer A: 8, 9, 15, 25, 20, 16, 24, 18, 21, 14, 16, 10
• mean = 16.3 ; standard deviation = 5.34

Dealer B: 7, 4, 10, 18, 21, 30, 27, 20, 16, 18, 12, 9

mean = 16; standard deviation = 7.6

On average, Dealer A sells more cars per month than Dealer B. Dealer A has a smaller standard deviation than Dealer B. Therefore, the amount of cars hat Dealer A sell from month to month varies less than that of Dealer B. That is, Dealer A is more consistent in the number of cars he sells than Dealer B.

Practice Problem #1

Directions: Find the Mean, Median, Mode, Range, IQR and Standard Deviation for the following data set.

• mean: 52.42
• median: 53
• mode: No Mode
• Range: 39
• IQR: 24
• SD: 12.62
If the average monthly temperature increased by 2 degrees each month, how would this affect the mean and standard deviation?
• The mean would increase by 2 and the standard deviation would remain the same.
If the average monthly temperature increased by 3 degrees in the Month of March and July, how would this change the mean?

The mean and the standard deviation would increase.

Practice Problem #2
• Directions:
• Find the mean value of the shutouts.
• Find the interquartile range.

mean: 72.8

IQR: 19

Practice Problem #3

Directions: The following is a list of lengths (in minutes) of 13 Movies. Find the Mean and Standard Deviation for the following data set.

90, 102, 120, 180, 90, 85, 90, 137, 120, 151, 97, 93, 120

mean: 113.46

SD: 27.46

Practice Problems # 4

The following is a list of lengths (in minutes) of 13 Movies:

90, 102, 120, 180, 90, 85, 90, 137, 120, 151, 97, 93, 120

If all movie times increased by 10 minutes, how would the mean and standard deviation be affected?

Mean would increase by 10 and the standard deviation would remain the same.

Assignment:
• Day 1 Handout
• Review packet