8/11/2019 Biostat - Descriptive Measures (Ungrouped Data)

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Math 600 Biostatistics

Handout: Descriptive Measures for Ungrouped Data

Measures of Central Tendency (Average): Mean, Median, and Mode

Measures of Other Tendency (Quantiles or Fractiles): Quartiles, Deciles, and Percentiles

Measures of Variation: Range, Standard Deviation, Average Deviation, Variance, and Coefficient of Variation

MEASURES OF CENTRAL TENDENCY

Mean (M or x

or ) Arithmetic Mean: the sum of entries divided by the number of entries.Median (Md or x~ ) the middle score of the entries, arranged in ascending or descending order.

Mode (Mo or x ) the entry or entries that appeared most frequently.

unimodal, bimodal, trimodal, multimodal data

Find the measures of central tendency of the ungrouped data:

a) 4 , 5 , 7 , 8 , 6 , 7 , 9

x = __________ x~ = __________ x = __________

b) 15 , 13 , 12 , 13 , 12 , 17 , 17 , 16

x = __________ x~ = __________ x = __________

MEASURES OF OTHER TENDENCY

- divisions of arranged data into four (quartiles), ten (deciles), or hundred (percentiles)

Given the same set of ungrouped data above, find the following:

a) 4 , 5 , 7 , 8 , 6 , 7 , 9

P75= _______________ Q1= _______________

b) 15 , 13 , 12 , 13 , 12 , 17 , 17 , 16

Q2= _______________ P43= _______________

MEASURES OF VARIATION

Range (R) - the difference of the highest score and the lowest score

R= HS - LS

Quartile Deviation (QD) - the average of the difference between the first and the third quartiles

Average Deviation (AD) - the average of the deviation of the scores from the mean

Standard Deviation (s or ) the measure of the average deviation in the data set

1n

x-xs

2

,

1n

n

xx

s

22

or 1)n(n

xxns

22

Variance (s2or

2) - the measure of the average squared difference between observed values and

their mean

Coefficient of Variation (CV) used to compares the variability of two groups

100%x

sCV

2

1Q 3Q

QD

N

xxAD

8/11/2019 Biostat - Descriptive Measures (Ungrouped Data)

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Given the ungrouped data 4 , 5 , 7 , 8 , 6 , 7 , 9,

Find the following:

a) Range

Range = HS LS = ____________b) Quartile Deviation

2

QQD 13

Q = ____________

c) Average Deviation

N

xxAD

= ____________

d) Standard Deviation

1n

x-xs

2

= ____________

1)n(n

xxns

22

= ____________

e) Variance = ____________

f) Coefficient of Variation = ____________

What measure of central tendency to use?MEAN

when data are close to each otherwhen data are interval / ratiowhen a unique data is wanted

when it is desired to find the most stable

measure of central tendency

when s, s2, ,or will be used later

MEDIAN

when data are skewed (having extreme values)

when data are ordinal

when it is desired to know the middle valueMODE

when a measure of central tendency is quickly

needed

when data are nominalwhen the most typical data is wanted

What measure of variation to use?RANGE

when data have extreme valuesQUARTILE DEVIATION

when median is used as measure of central

tendencyAVERAGE DEVIATION

when it is desired to know the average of the

scores from meanSTANDARD DEVIATION / VARIANCE

when mean is used as measure of central

tendency

when it is desired to use a measure ofvariability that is most stable

when an estimation of percentage fallingbetween two scores is wanted

COEFFICIENT OF VARIATION

when it is desired to know which set of datahaving different means, is more varied