Measures of Central Tendency

Chapter 3.2 – Tools for Analyzing Data

Mathematics of Data Management (Nelson)

MDM 4U

Sigma Notation the sigma notation is used to compactly

express a mathematical series ex: 1 + 2 + 3 + 4 + … + 15 this can be expressed:

the variable k is called the index of summation.

the number 1 is the lower limit and the number 15 is the upper limit

we would say: “the sum of k for k = 1 to k = 15”

15

1k

k

Example 1:

write in expanded form:

This is the sum of the term 2n+1 as n takes on the values from 4 to 7.

= (2×4 + 1) + (2×5 + 1) + (2×6 + 1) + (2×7 + 1) = 9 + 11 + 13 + 15 = 48 NOTE: any letter can be used for the index of

summation, though a, n, i, j, k & x are the most common

7

4

)12(n

n

Example 2: write the following in sigma notation

8

3

4

3

2

33

3

0 2

3

nn

The Mean

n

xx

n

ii

1

Found by dividing the sum of all the data points by the number of elements of data

Affected greatly by outliers Deviation

the distance of a data point from the mean calculated by subtracting the mean from the value i.e. xx

The Weighted Mean

n

ii

n

iii

w

wxx

1

1

where xi represent the data points, wi represents the weight or the frequency

“The sum of the products of each item and its weight divided by the sum of the weights”

see examples on page 153 and 154 example: 7 students have a mark of 70 and 10 students

have a mark of 80 mean = (70×7 + 80×10) ÷ (7+10) = 75.9

Means with grouped data

for data that is already grouped into class intervals (assuming you do not have the original data), you must use the midpoint of each class to estimate the weighted mean

see the example on page 154-5 and today’s Example 4

Median

the midpoint of the data calculated by placing all the values in order if there is an odd number of values, the median is

the middle number 1 4 6 8 9 median = 6

if there are an even number of values, the median is the mean of the middle two numbers 1 4 6 8 9 12 median = 7

not affected greatly by outliers

Mode

The number that occurs most often There may be no mode, one mode, two modes (bimodal), etc. Which distributions from yesterday have one mode? Mound-shaped, Left/Right-Skewed Two modes? U-Shaped, some Symmetric Modes are appropriate for discrete data or non-numerical data

Eye colour Favourite Subject

Distributions and Central Tendancy the relationship between the three measures

changes depending on the spread of the data

symmetric (mound shaped) mean = median = mode

right skewed mean > median > mode

left skewed mean < median < mode

Co

un

t

1

2

3

data0 1 2 3 4 5 6 7

Data Histogram

Co

un

t

1

2

3

4

5

data0 1 2 3 4 5 6 7

Data Histogram

Co

un

t1

2

3

4

5

data0 1 2 3 4 5 6 7

Data Histogram

What Method is Most Appropriate? Outliers are data points that are quite

different from the other points Outliers affect the mean the greatest The median is least affected by outliers Skewed data is best represented by the

median If symmetric either median or mean If not numeric or if the frequency is the most

critical measure, use the mode

Example 3 a) Find the mean, median and mode

mean = [(1x2) + (2x8) + (3x14) + (4x3)] / 27 = 2.7 median = 3 (27 data points, so #14 falls in bin 3) mode = 3

b) What shape does it have? Left-skewed

Survey responses 1 2 3 4

Frequency 2 8 14 3

Example 4 Find the mean, median and mode

mean = [(145.5×3) + (155.5×7) + (165.5×4)] ÷ 14

= 156.2 median = 151-160 or 155.5 mode = 151-160 or 155.5

MSIP / Homework: p. 159 #4, 5, 6, 8, 10-13

Height 141-150 151-160 161-170

No. of Students 3 7 4

MSIP / Homework

p. 159 #4, 5, 6, 8, 10-13

References

Wikipedia (2004). Online Encyclopedia. Retrieved September 1, 2004 from http://en.wikipedia.org/wiki/Main_Page