Coverage• Measures of Central Tendency– Mean– Median– Mode

• Measures of Variability and Dispersion– Range– Average deviation– Variance– Standard deviation

Introduction to Notations

If variable X is the variable of interest, and that n measurements are taken;

then the notation X1, X2, X3, … , Xn will be used to represent n observations.

Sigma

Indicates “summation of”

Σ

Summation Notation

If variable X is the variable of interest, and that n measurements are taken;

the sum of n observations can be written as

Σ Xi = X1+X2+ … +Xn

n

i=1

Summation Notation

Σ Xi = X1+X2+ … +Xn

n

i=1

Upper limit of summation

Lower limit of summation

Greek letter Sigma

Rules of Summation

summation of the sum of variables is

Σ (Xi+Yi) = Σ Xi+ Σ Yi n

i=1

n

i=1

n

i=1

the sum of their summations

Σ (ai+bi+…+zi) =

Σ ai+ Σ bi + … + Σ zi

n

i=1

n

i=1

n

i=1

n

i=1

The summation of the sum of variables is…

the sum of their summations.

Rules of Summation

Σ cXi = c Σ Xi

= c(X1+X2+ … +Xn)

n

i=1

n

i=1

If c is a constant, then…

Rules of Summation

Σ c = ncn

i=1

The summation of a constant is the product of upper limit of summation n and constant c.

MEASURES OF CENTRAL TENDENCYStatistics in Research

Mean

• The sum of all values of the observations divided by the total number of observations

• The sum of all scores divided by the total frequency

Population mean μ = Σ Xi

N

i=1

N

Sample mean x = Σ Xi

n

i=1

n

Σ fiXi

n

i=1

n

Mean in an Ungrouped Frequency

= (f1X1+f2X2+ … +fnXn)

where f is the frequencyof the occurring score

n

Properties - Mean

• The most stable measure of central tendency• Can be affected by extreme values• Its value may not be an actual value in the

data set• If a constant c is added/substracted to all

values, the new mean will increase/decrease by the same amount c

Median

• Positional middle of an array of data• Divides ranked values into halves with 50%

larger than and 50% smaller than the median value.

If n is odd:

Md = X(n+1)/2

If n is even:

Md = Xn/2 + X(n/2)+1

2

Properties - Median

• The median is a positional measure• Can be determined only if arranged in order• Its value may not be an actual value in the

data set• It is affected by the position of items in the

series but not by the value of each item• Affected less by extreme values

Mode

• Value that occurs most frequently in the data set

• Locates the point where scores occur with the greatest density

• Less popular compared to mean and median measures

Properties - Mode

• It may not exist, or if it does, it may not be unique

• Not affected by extreme values• Applicable for both qualitative and

quantitative data

MEASURES OF VARIABILITY AND DISPERSION

Statistics in Research

Range

• Measure of distance along the number line over where data exists

• Exclusive and inclusive range– Exclusive range = largest score - smallest score– Inclusive range = upper limit - lower limit

Properties - Range

• Rough and general measure of dispersion• Largest and smallest extreme values

determine the range• Does not describe distribution of values within

the upper and lower extremes• Does not depend on number of data

Absolute Deviation

Average of absolute deviations of scores from the mean (Mean Deviation) or the median (Median Absolute

Deviation)

MD = Σ | Xi - X |n

i=1

n

MAD = Σ | Xi - Md |n

i=1

n

Properties – Absolute Deviation

• Measures variability of values in the data set• Indicates how compact the group is on a

certain measure

Variance

• Average of the square of deviations measured from the mean

• Population variance (σ2) and sample variance (s2)

σ2 = Σ ( Xi - μ )2N

i=1

N

s2 = Σ ( Xi – X )2n

i=1

n -1

s2 = nΣ Xi2 - (Σ Xi )2

n

i=1

n(n -1)

n

i=1

Properties – Variance

• Addition/subtraction of a constant c to each score will not change the variance of the scores

• Multiplying each score by a constant c changes the variance, resulting in a new variance multiplied by c2

Standard Deviation

• Square root of the average of the square of deviations measured from the mean – square root of the variance

• Population standard deviation (σ) and sample standard deviation (s)

σ = Σ ( Xi - μ )2N

i=1

N

s = Σ ( Xi – X )2n

i=1

n -1

Why n-1?• Degrees of freedom– Measure of how much precision an estimate of

variation has– General rule is that the degrees of freedom

decrease as more parameters have to be estimated

• Xbar estimates μ• Using an estimated mean to find the standard

deviation causes the loss of ONE degree of freedom

Properties – Standard Deviation

• Most used measure of variability• Affected by every value of every observation• Less affected by fluctuations and extreme

values

Properties – Standard Deviation

• Addition/subtraction of a constant c to each score will not change the standard of the scores

• Multiplying each score by a constant c changes the standard deviation, resulting in a new standard deviation multiplied by c

Choosing a measure

• Range– Data are too little or scattered to justify more

precise and laborious measures– Need to know only the total spread of scores

• Absolute Deviation– Find and weigh deviations from the mean/median– Extreme values unduly skews the standard

deviation

Choosing a measure

• Standard Deviation– Need a measure with the best stability– Effect of extreme values have been deemed

acceptable– Compare and correlate with other data sets

FREQUENCY DISTRIBUTIONStatistics in Research

74 79 69 72 53 76 62 82 84 87 9672 79 68 71 50 75 60 81 84 86 9172 77 66 69 50 75 59 80 82 85 8872 77 66 69 50 75 60 81 83 85 8973 78 68 70 50 75 60 81 83 86 8973 59 65 69 50 75 77 80 82 84 8773 79 68 71 51 76 62 81 84 87 9273 79 68 71 52 76 62 82 84 87 9474 79 68 71 53 76 62 82 84 87 9450 57 63 69 72 74 77 80 82 84 87

Raw data

50 57 63 69 72 74 77 80 82 84 8750 59 65 69 72 75 77 80 82 84 8750 59 66 69 72 75 77 80 82 85 8850 60 66 69 72 75 77 81 83 85 8950 60 68 70 73 75 78 81 83 86 8950 60 68 71 73 75 79 81 84 86 9151 62 68 71 73 76 79 81 84 87 9252 62 68 71 73 76 79 82 84 87 9453 62 68 71 74 76 79 82 84 87 9453 62 69 72 74 76 79 82 84 87 96

Array

Frequency Distribution Table

• Class Frequency– Number of observations within a class, f

• Class Limits– End numbers of the class

• Class Interval– Interval between the upper and lower class limits,

ie: [Xupper limit , Xlower limit ]

Frequency Distribution Table• Class Boundaries– True limits of the class, halfway between class limit of

the current class and that of the preceding/succeeding class, LCB and UCB

• Class Size– Difference between UCB and LCB,

ie: XUCB - XLCB

• Class Mark– Midpoint of the class interval, average value of the

upper and lower class limits, ie. Xupper limit - Xlower limit

Constructing an FDT

• Determine number of classes– Sturges Formula, K = 1 + 3.322 log n– Square Root, K = sqrt(n)

• Determine the approximate class size, C’ = R/K• Round off C’ to a more convenient number C

Constructing an FDT

• Determine lower class limit– Lowest class should not be empty, must contain

the lowest value in the data set

• Determine succeeding lower class limits by adding class size C to the current lower class limit

• Tally frequencies

50 57 63 69 72 74 77 80 82 84 8750 59 65 69 72 75 77 80 82 84 8750 59 66 69 72 75 77 80 82 85 8850 60 66 69 72 75 77 81 83 85 8950 60 68 70 73 75 78 81 83 86 8950 60 68 71 73 75 79 81 84 86 9151 62 68 71 73 76 79 81 84 87 9252 62 68 71 73 76 79 82 84 87 9453 62 68 71 74 76 79 82 84 87 9453 62 69 72 74 76 79 82 84 87 96

Array

Frequency Distribution TableClass Frequency LCB UCB RF <CF >CF

50-54 10 49.5 54.5 0.09 10 11055-59 3 54.5 59.5 0.03 13 10060-64 8 59.5 64.5 0.07 21 9765-69 13 64.5 69.5 0.12 34 8970-74 17 69.5 74.5 0.15 51 7675-79 19 74.5 79.5 0.17 70 5980-84 22 79.5 84.5 0.20 92 4085-89 13 84.5 89.5 0.12 105 1890-94 4 89.5 94.5 0.04 109 595-99 1 94.5 99.5 0.01 110 1

Other Terms• Relative frequency, RF– Class frequency divided by number of

observations, ie. RF = fi / n• Relative Frequency Percentage, RFP– RF = (fi / n) x 100%

• Cummulative frequency– Shows accumulated frequencies of successive

classes, either from the beginning (less than CF) or end (greater than CF) of the FDT

Mean from an FD

X = Σ fiXi

K

i=1

Σ fi

K

i=1

where Xi = class mark of the ith class

Median from an FD

Md = LCBMd + C n/2 - <CFMd-1

where LCBMd = lower class boundary of median class <CFMd-1 = less than cumulative frequency preceeding the

median class

fMd

Mode from an FD

Mo = LCBMo + C fMo - fMo-1

where LCBMo = lower class boundary of modal classfMo, fMo-1, fMo+1 = frequency of modal class, class preceding and

class succeeding the modal class

2fMo - fMo-1 - fMo+1

MD = Σ fi |Xi - X|n

i=1

n

Mean Deviation from an FD

where Xi = class mark of the ith class n = total number of observations; total

frequency, ie. n = Σ fi

s2 = Σ fi(Xi - X)2n

i=1

(n -1)

Variance from an FD

where Xi = class mark of the ith class n = total number of observations; total

frequency, ie. n = Σ fi

s2 = nΣ fiXi2 - (Σ fiXi )2

n

i=1

n(n -1)

Variance from an FD

n

i=1

where Xi = class mark of the ith class n = total number of observations; total

frequency, ie. n = Σ fi