Working with one variable data

Measures of Central TendencyIn statistics, the three most commonly used measures of central tendency are:MeanMedianMode

Each measure has its particular advantage and disadvantage for a given set of data.

MeanMost commonly referred to as the

averageTo find the mean, add up all of the

numbers in your list and divide by the number of numbers.

Really good when the data is fairly close together.

Most commonly used.

MeanIn statistics, it is important to distinguish between the mean of a population and the mean of a sample of that population

Mean - Population

Nx

Nxxx N

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The Greek letter mu, μ – Represents a Population Mean

∑ x – is the sum of all values of X in the population.

N – is the number of values in the entire population.

Mean - Samplex read as “ x-bar – Represents a Sample Mean

– is the sum of all values of X in the population.

N – is the number of values in the entire population.

nx

nxxxx n

21

x

MedianThe median is the middle entry in

an ordered list. There are as many data points above it as below it.

When there is an even number of values, the median is the midpoint between the two middle values.

ModeThe mode is the most frequent number in a

data set.

There can be no mode as well as more than one mode.

Good when the value of the number is the most important information (e.g. shoe size).

Only choice with categorical data.

OutliersValues distant from the majority of the

data.The median is often a better measure

of central tendency than the mean for small data sets that contain outliers.

For larger data sets, the effect of outliers on the mean is less significant.

Choosing a Measure of Central TendencyIf data contains outliers, use the

medianIf the data are strongly skewed , use

medianIf data is roughly symmetrical, the

mean and the median will be close, so either is appropriate.

If data is not numeric, use the mode.

ExampleThe physics exam had the following

results.71, 82, 55, 76, 66, 71, 90, 84, 90,

64, 71, 70, 83, 45, 73, 51 68Determine the mean, median, and

mode.

Example - MeanThe physics exam had the following results.

71, 82, 55, 76, 66, 71, 90, 84, 90, 64, 71, 70, 83, 45, 73, 51 68

17685173458370716490716676558271

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Nxxx

Nx N

171215

5.71

Example - MedianThe physics exam had the following results.

71, 82, 55, 76, 66, 71, 90, 84, 90, 64, 71, 70, 83, 45, 73, 51 68

Order the data: 45, 51, 55, 64, 66, 68, 70, 71, 71, 71, 73, 76, 82, 83, 84, 90, 90,

Therefore the median is 71.

Example - ModeThe physics exam had the following results.

71, 82, 55, 76, 66, 71, 90, 84, 90, 64, 71, 70, 83, 45, 73, 51 68

Therefore the mode is 71.

Weighted Mean•Sometimes, certain data within a set are more significant than others.

•A weighted mean gives a measure of central tendency that reflects importance of the data

•Weighted means are often used in calculations of indices

wx

Weighted Mean

ii

iii

n

nnw

w

xw

wwwxwxwxwx

21

2211

– sum of the weighted values.

– sum of the various weighting factors.

i

iixw

i

iw

Weighted Mean - ExampleThe averages (means) of five Data

Management classes are 69, 72, 66, 75, and 78. If the class sizes were 26, 33, 25, 35, and 37 respectively, determine the overall

average (mean) for the entire grade.

Weighted Mean - ExampleClass Mean, xi

Weight FactorClass Size, wi

1 69 26

2 72 33

3 66 25

4 75 35

5 78 37

Weighted Mean

373525332678377535662572336926

ii

iii

w w

xwx

15633111

6.72The average for the

entire grade is 72.6%

Mean for Group Data•The mean should always be calculated using the original data before they are grouped into intervals.

•If you are presented with the data already summarized in a frequency table approximation of the centres of the data can be made.

Mean for Group Data

ii

iii

f

mf

ii

iii

f

mfx

– sum of the interval midpoints times the number of data in the interval.

– sum of all the frequencies.

i

iimf

i

if

Mean for Group Data - ExampleThe following table represents the number of hours per day of watching TV in a sample of 500 people.

Numberof hours 0-1 2-3 4-5 6-7 8-9 10-11 12-13

Frequency 64 92 141 86 71 35 11

a) What is the mean number of TV viewing hours in this group?

b) What length of time is most often spent in front of a TV by this group?

c) What is the median number of TV viewing hours?

IntervalMidpoint

(mi)Frequency

fi

Cumulative Frequency

fixi

0 - 1 0.5 64 64 64 x 0.5 = 32.0

2 - 3 2.5 92 156 92 x 2.5 = 230.0

4 - 5 4.5 141 297 141 x 4.5 = 634.5

6 -7 6.5 86 383 86 x 6.5 = 559.0

8 - 9 8.5 71 454 71 x 8.5 = 603.5

10 - 11 10.5 35 489 35 x 10.5 = 367.5

12 - 13 12.5 11 500 11 x 12.5 = 137.5

500i

if 0.2564 ii

imf

1. Find the midpoints and cumulative frequencies for the intervals

2. Calculate the midpoints times frequency for each interval

3. Determine the sum of frequency and fimi

Mean for Group Data

ii

iii

f

mf.

5002564.

1.5.

The mean number of viewing hours for this group was approximately five hours.

Mean for Group Data - Exampleb) What length of time is most often spent in front of a

TV by this group?The mode is the answer to this question. From the frequency table the model interval is identified by

the larges frequency.

The most frequent period of time spent in front of a TV by this group is between four and five hours.

Mean for Group Data - Examplec) What is the median number of TV viewing hours?

The median is the middlemost datum. The median is the average of the 250th or 251th By referring to

the cumulative frequency column we notice that the 250th or 251th data occur in the interval 4-5.

We would then estimate the median to be 4.5 hours of viewing time.

Homework

Pg 133 #1,3,5,7,8,9,11