# Measures of Central Tendency and

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```					Educational Statistics

Measures of Central Tendency
and Variability (Dispersion)
Over the Counter Drug Sales
(in Millions)
Tylenol      855          Mylanta       135
Advil       360           Tums         135
Vicks       350         Excedrin       130
One Touch      220         Benadryl       130
Robitussin    205           Halls        130
Bayer       170         Metamucil      125
Alka-Seltzer   160          Sudafed       115
Centrum      150
What is the AVERAGE number of drug sales?
Measures of Central Tendency

Measures of central tendency tell us
something about the “typicalness” of a set
of data.
• Tell us what the typical score is in a
distribution of scores.
• Three measures of central tendency:
– Mode
– Median
– Mean
Measures of central tendency:
the mode
The mode is the score that occurs most
frequently in a distribution.
Sometimes more than one score occurs at
frequencies distinctively higher than other
scores, in which case there is(are) more
than one mode:
– Bi-modal distributions.
– Multi-modal distributions.
Only appropriate measure of central
tendency for nominal data.
Over the Counter Drug Sales
(in Millions)

Tylenol       855            Mylanta        135
Advil        360             Tums          135
Vicks        350            Excedrin       130
One Touch       220            Benadryl       130
Robitussin     205              Halls        130
Bayer        170           Metamucil       125
Alka-Seltzer    160            Sudafed        115
Centrum       150
What is the MODE of this distribution?
Measures of central tendency:
the median (Md)
The median is the middle score in an ordered
distribution of scores.
It is the score that divides a distribution in
half.
It is also the score at the 50th percentile
rank.
The median can be found by computing the
median location: (N + 1)/2.
The median is the most appropriate measure
of central tendency for ordinal data.
Measures of central tendency:
the median (Md)
In the distribution given   Score Freq Cum. F %c
to the right, find the     6    2      2     3.8
median.                    8    5      7    13.5
What is the percentile       9    0      7    13.5

rank (PR) of a score of   10    8     15    28.8

13?
11   11     26    50.0
12    9     35    67.3
What is the score cor-      13    6     41    78.8
responding to a           14   4      45    86.6
percentile rank of 29?    15    5     50    96.2
16    2     52   100.0
Over the Counter Drug Sales
(in Millions)
Tylenol       855              Mylanta       135
Advil        360               Tums         135
Vicks        350             Excedrin       130
One Touch       220             Benadryl       130
Robitussin     205               Halls        130
Bayer        170             Metamucil      125
Alka-Seltzer    160              Sudafed        115
Centrum       150
What is the Median of this distribution?
Measures of central tendency:
the Mean (µ,M, or     )

• The most widely used measure of
central tendency:

• In words, the mean ( ) is the sum (Σ)
of the scores (the X’s) divided by the
number of scores (N).
Over the Counter Drug Sales
(in Millions)
Tylenol       855              Mylanta         135
Advil        360                Tums          135
Vicks        350              Excedrin        130
One Touch       220              Benadryl        130
Robitussin     205                Halls         130
Bayer        170             Metamucil        125
Alka-Seltzer    160               Sudafed        115
Centrum       150
What is the Mean of this distribution?
Measures of Central Tendency
with special Distributions
• The mode and bimodal distributions.
– For distributions with more than one mode, the
other measures of central tendency are
• The Median and skewed distributions.
– When a distribution is skewed the use of the
mean may be misleading
– Skew can be determined by the relative
positions of the mean, median, and mode.
Measures of Variability
• How would you describe the   410   500
variability in the
distribution of SAT-V        450   515
scores given at the right?
• In other words, how          465   535
scores?                      485   545

• Think about it.              500   585
• Write these values down.
Measures of Variability
• Three common measures of variability
are
– The Range.
– The Variance.
– The Standard deviation.
• Other measures of variability are
– The interquartile range.
– The quartile deviation or semi-interquartile
range.
Measures of Variability
The Range: (R)
• R = The difference between the
largest value in the distribution and
the smallest value in the distribution.
• I.e. R = Xlargest – Xsmallest.
• Compute the Range for the
distribution given.
• R = 175.
Measures of Variability
The Variance (Var):
• The variance is more computationally
comples.
• Defined as the average squared
deviation from the mean of the
distribution.
• In symbols:
Computing the Variance

• First, compute   X     X
the sum:
410   500
450   515
465   535
485   545
500   585
Computing the Variance

• First, compute    X     X
the sum:
410   500
450   515
• Then, divide by   465   535
N:                485   545
500   585
Computing the Variance
• Next, subtract the X      d   X     d
mean from each
score (call these   410       500
deviations from the
mean, or, d ):      450       515

465       535

485       545

500       585
Computing the Variance
• Next, subtract the X      d     X     d
mean from each
score (call these   410   -89   500   1
deviations from the
mean, or, d ):      450   -49   515   16

465   -34   535   36

485   -14   545   46

500   1     585   86
Computing the Variance
• Next, Square the      d     d2   d    d2
deviations from the
-89        1
mean:
-49        16

-34        36

-14        46

1          86
Computing the Variance
• Next, Square the      d     d2     d    d2
deviations from the
-89   7921   1     1
mean:
-49   2401   16   256

-34   1156   36   1296

-14   196    46   2116

1      1     86   7396
Computing the Variance
Now, sum the           d     d2     d    d2
squared deviations:
-89   7921   1     1

-49   2401   16   256

-34   1156   36   1296
And divide by N:
-14   196    46   2116

1      1     86   7396
Measures of Variability
The Standard Deviation:
Generally, we would prefer a measure of
variability that tells us something about
how far, on average, scores deviate from
the mean.
This is what the standard deviation tells us.
Since the variance is the average squared
deviation from the mean, the standard
deviation, computed as the square root of
the variance gives us the average deviation
from the mean.
Measures of Variability
The Coefficient of Variation (CV)
Distributions with larger means tend to
have larger variances (and SDs) than
distributions with smaller means.
The CV provides convenient way to
compare the variances of two or more
distributions.
SD
CV 
X
Using Statistics as Estimators
We are rarely interested in sample
statistics…we are interested in
population parameters.
Statistics are used to estimate (or
make inferences about) parameters.
The best statistics are sufficient,
unbiased, efficient, and robust (or
resistant)
End of Presentation

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