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.

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:

1 2 3 ... nx x x xx

n

x

Example

Find the mean of the set of data.

75, 100, 63, 95, 82, 78

75+100 63+95+82+78

6

82.1667

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)

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

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

78 is the median

Example

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

63, 75, 78, 82, 95, 100

82, 63, 100, 75, 78, 95

78 82

2

+

80 is the median

You try: Find the median

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

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

12median

2.5median

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 mode1. Find the mode of the data set: 1, 2, 3, 4, 5, 3

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

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

3 is the mode

There is no mode

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:

1. Find the range of the data set: 63, 75, 78, 82, 95, 100 100 - 63 = 37

2. 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

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

Find the medianFind the 1st and 3rd quartiles

IQR = 95 - 63 = 32

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

You Try:

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

Find the medianFind the 1st and 3rd quartiles

IQR = 91 - 78 = 13

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

Example

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

IQR 97.5 69

28.5

63 752

95 1002

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

69 97.5

You Try:

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

IQR 5.5 2 3.5

5 6

2

2 2

2

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

2 5.5

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

(xi )2

i1

n

n

xxS

Example:

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

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

4.5607x

3.1145xS

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?

1.25xS

Use xS

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?

9.52x

Use x

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?

3.54xS

Use xS

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?

1.77x

Use x

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, 9mean = 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: 72.8IQR: 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