Measures of Central Tendency

Section 5.1

Standard: MM2D1 bc

Essential Question: How do I calculate and use measures of central tendency and dispersion to compare data sets?

Statistics: numerical values used to summarize and compare sets of data

Mean: average, denoted by

Median: the middle number when the numbers are in order

Mode: the number or numbers that occur more frequently

Measures of dispersion: a statistic that tells you how spread out the data values are

Range: the difference between the greatest and least data values

Standard deviation: the measure that describes the typical difference between a data value and the mean, denoted by σ

Interquartile range: the difference between the upper and lower quartile of a set of data

Variance: standard deviation squared.

x

Find the following.

1. 42, 78, 56, 95, 49, 55, 63, 71

Mean: _______

Median: _______

Mode: _______

Range: _______

Q1: _______

Q2: _______

Q3: _______

IQR: _______

42 + 78 + 56 + 95 + 49 + 55 + 63 + 71 = 509 509/8 63.63

63.63 59.5

42, 49, 55, 56, 63, 71, 78, 95

Median(56 + 63)/2

59.5Q2

none 95 – 4253

Q1

(49 + 55)/252

52 59.5

Q3

(78 + 71)/274.5

74.5 74.5 – 5222.5

Standard deviation of population σ (read as “sigma”) of x1, x2, …, xn is:

Now find the standard deviation of example 1: 42, 78, 56, 95, 49, 55, 63, 71

2 2 2

1 2 ... nx x x x x x

n

2 2 2 2

2 2 2 2

42 63.625 78 63.625 56 63.625 95 63.625

49 63.625 55 63.625 63 63.625 71 63.625

8

2 2 2 2

2 2 2 2

21.625 14.375 7.625 31.375

14.625 8.625 0.625 7.3755

8

467.64 206.64 58.14 984.39 213.89 74.39 0.39 54.40

8

2059.38

8

275.42

16.04

Standard Deviation on the Calculator:

2nd DATA ENTER (1-VAR)DATA• Input data into x1, x2, x3, …• Use to move to next data entry• Notice there is a FRQ after each xn term. This is means

frequency. If a number appears in the list more than once you may change this to the number of times the number appears in the list.

STATVAR• n = number of terms • = mean• Sx = sample stand. dev. • σx = population stand. dev.• Σx = sum of data • Σx2 = sum of data squared

x

2. 15, 11, 15, 14, 14, 13, 17

Mean: _______

Median: _______

Mode: _______

Range: _______

Q1: _______

Q2: _______

Q3: _______

IQR: _______

Standard deviation: _______

Variance: ______

15 + 11 + 15 + 14 + 14 + 13 + 17 = 99 99/7 = 14.14

14.14 14

11, 13, 14, 14, 15, 15, 17

Median14Q2

14, 15 17 – 116

Q1

13 13 14

Q3

15

15 15 – 132

1.73 2.98

(Standard deviation)2

Draw a box and whisker plot for the data set.

Now check for outliers. (Remember: 1.5 • IQR)

10 11 12 13 14 15 16 17 18 19 20

1.5(2) = 3Q3 + 3 = 15 + 3 = 18 (No values larger than 18)Q1 – 2 = 13 – 3 = 10 (No values smaller than 10)Therefore, no outliers.

3. Compare the means and standard deviations of Set A and Set B.

Set A mean: _______ Set B mean: _______

standard deviation: _______ standard deviation: _______

Set A 7 3 4 9 2

Set B 5 8 7 6 4

5

2.6

6

1.4

Set B has a higher average and Set A’s data is more spread out.

4. The lists show the number of cars sold each month for one year by two competing car dealers. Compare the mean and standard deviation for the numbers of cars sold by the two car dealers.

Dealer A: 8, 9, 15, 25, 28, 16, 24, 18, 21, 14, 16, 10Dealer B: 7, 4, 10, 18, 21, 30, 27, 20, 16, 18, 12, 9

Dealer A mean: _______ Dealer B mean: _______

standard deviation: _______ standard deviation: _______

17

6.2

16

7.6

Dealer A sells more cars on average and Dealer B’s data is more spread out.