The mean is the 'center of gravity' of a distribution of scores. It is the point on which
the distribution would balance.
The Greek letter μ (mu) is used to indicate the population mean. In general, Greek letters are used to represent parameters (population values), although there are exceptions to this convention.
The sum of squared deviations about the sample mean is a minimum, i.e., &Sigma(X-
)2 is
a minimum (see proof).
If a distribution has an odd number of values, then the midpoint or median is the point where ther are an equal number of values above and below.
median = 20 (since two values fall above and two values fall below 20)
If a distribution has an even number of values, then the median is the point half way between to two middle-most values, i.e., just average the two middle-most values.
median = 21 (since that point is half way between the values 22 and 20)
The most important point about finding the median is that the values must be sorted before you can determine the median.
Example: [1 2 3 4 45] mean = 11 median = 3
mode = 26 median = 29 mean = 27.9
use http://www.philender.com/courses/data/hsb2, clear
summarize read write math science socst
Variable | Obs Mean Std. Dev. Min Max
---------+-----------------------------------------------------
read | 200 52.23 10.25294 28 76
write | 200 52.775 9.478586 31 67
math | 200 52.645 9.368448 33 75
science | 200 51.85 9.900891 26 74
socst | 200 52.405 10.73579 26 71
summarize write, detail
writing score
-------------------------------------------------------------
Percentiles Smallest
1% 31 31
5% 35.5 31
10% 39 31 Obs 200
25% 45.5 31 Sum of Wgt. 200
50% 54 Mean 52.775
Largest Std. Dev. 9.478586
75% 60 67
90% 65 67 Variance 89.84359
95% 65 67 Skewness -.4784158
99% 67 67 Kurtosis 2.238527
sort female
by female: summ write, detail
-> female= male
writing score
-------------------------------------------------------------
Percentiles Smallest
1% 31 31
5% 33 31
10% 37 31 Obs 91
25% 41 31 Sum of Wgt. 91
50% 52 Mean 50.12088
Largest Std. Dev. 10.30516
75% 59 65
90% 63 65 Variance 106.1963
95% 65 67 Skewness -.179898
99% 67 67 Kurtosis 1.872877
-> female= female
writing score
-------------------------------------------------------------
Percentiles Smallest
1% 35 35
5% 41 35
10% 44 36 Obs 109
25% 50 37 Sum of Wgt. 109
50% 57 Mean 54.99083
Largest Std. Dev. 8.133715
75% 62 67
90% 65 67 Variance 66.15732
95% 65 67 Skewness -.5899993
99% 67 67 Kurtosis 2.544105
table female, contents(freq mean write median write sd write)
----------+---------------------------------------------------
female | Freq. mean(write) med(write) sd(write)
----------+---------------------------------------------------
male | 91 50.12088 52 10.30516
female | 109 54.99083 57 8.133716
----------+---------------------------------------------------
tabstat write, by(female) stat(n mean median sd)
Summary for variables: write
by categories of: female
female | N mean p50 sd
-------+-----------------------------------------
male | 91 50.12088 52 10.30516
female | 109 54.99083 57 8.133715
-------+-----------------------------------------
Total | 200 52.775 54 9.478586
-------------------------------------------------
univar read write math science socst
/* univar is not built into Stata */
/* findit univar */
-------------- Quantiles --------------
Variable n Mean S.D. Min .25 Mdn .75 Max
-------------------------------------------------------------------------------
read 200 52.23 10.25 28.00 44.00 50.00 60.00 76.00
write 200 52.77 9.48 31.00 45.50 54.00 60.00 67.00
math 200 52.65 9.37 33.00 45.00 52.00 59.00 75.00
science 200 51.85 9.90 26.00 44.00 53.00 58.00 74.00
socst 200 52.41 10.74 26.00 46.00 52.00 61.00 71.00
-------------------------------------------------------------------------------
univar write, by(female)
-> female=male
-------------- Quantiles --------------
Variable n Mean S.D. Min .25 Mdn .75 Max
-------------------------------------------------------------------------------
write 91 50.12 10.31 31.00 41.00 52.00 59.00 67.00
-------------------------------------------------------------------------------
-> female=female
-------------- Quantiles --------------
Variable n Mean S.D. Min .25 Mdn .75 Max
-------------------------------------------------------------------------------
write 109 54.99 8.13 35.00 50.00 57.00 62.00 67.00
-------------------------------------------------------------------------------
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Phil Ender, 30Jun98