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Practice
Descriptive statisticsTablesGraphs
2920 4050 2300 3400 3912 3234 3110 4045 3310 2594 2187 2700 3300 4250 3350 2750 2342 2350 4680 4123 3470 2084 2211 2450 4070 4360 3700 2495 3203 2950 3250 2920 2630 2466 3100 2780 4265 1588 3700 4990 3510 3020 4174 2950 3380 2607 3225 2410 3450 1800 3110 3860 4100 2540 2960 1790 3610 2225 3232 2625
Birthweights of 60 infants are given below:
Find
Mean, median and mode
Q1, Q3
Standard deviation and coefficient of variation
60
2625254031102920
60n
60
1 i
n
1 ixx ii
2.313660
188167x
Average birthweight of infants is approximately 3136 gr.
Arithmetic Mean
x
1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990
Ordered data are:
Median
30th value
31st value
1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990
n=60
5.30261
2160
21
n
i Median is 3110
Multiple modes exist.
Mode
valuethQ 25.154
1601
1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990
The difference between 15th and 16th observations is 54.
54/4=13.5
Q1=2540+13.5=2553.5
We add 13.5 to 15th observation to find Q1
valuethQ 75.454
)160(33
1588 2410 2750 3110 3400 4050 1790 2450 2780 3203 3450 4070 1800 2466 2920 3225 3470 4100 2084 2495 2920 3232 3510 4123 2187 2540 2950 3234 3610 4174 2211 2594 2950 3250 3700 4250 2225 2607 2960 3300 3700 4265 2300 2625 3020 3310 3860 4360 2342 2630 3100 3350 3912 4680 2350 2700 3110 3380 4045 4990
Q3=3610+67.5=3677.5
The difference between 45th and 46th observations is 90.
3*(90/4)=67.5
We add 67.5 to 45th observation to find Q3
1
2
2
nn
xx
s
ii
3623,689,082625254031102920 22222 ix
188,1672625254031102920 ix
,88935,406,819)( 2 ix
grs 754.37160
60/889,819,406,35083,689,623
Average distance of observations from the mean is 754.37 gr.
If infant weight were given as kg instead of gr, what would be the result of the mean and standart deviation.
Find the result without using any formulations.
0.75437(1,000)754.37/s754.37
1362.3000,1/2.136,32.136,3
2
s
xx
%241003136
37.754100
x
sCV
The amount variation between observations in this distribution is 24% in terms of the mean.
L=1588
H=4990
Range=H-L=4990-1588=3402
#of classes=8
Class Int.=3402/8=425.25426
To find lower limits:
1588
1588+426=2014
2014+426=2440...4114+426=4570
To find upper limits:
2014-1=2013
2013+426=2439
2439+426=2865...4569+426=4995
Lower-upper limit
1588-2013
2014-2439
2440-2865 . . . 4570-4995
CLASSES Lower Limit Upper Limit Frequency bi
Relative Frequency
Class Mid-Point
True Class Limit
Less than ... Frequency
Less than ... Rel. Freq.
1587.5 0 0.00
1588 2013 3 -3 5.0 1800.5 2013.5 3 5.00
2014 2439 8 -2 13.33 2226.5 2439.5 11 18.33
2440 2865 11 -1 18.33 2652.5 2865.5 22 36.67
2866 3291 14 0 23.33 3078.5 3291.5 36 60.00
3292 3717 11 1 18.33 3504.5 3717.5 47 78.33
3718 4143 7 2 11.67 3930.5 4143.5 54 90.00
4144 4569 4 3 6.67 4356.5 4569.5 58 96.67
4570 4995 2 4 3.33 4756.5 TOTAL 60 100.00 4995.5 60 100.00
Mean
4.314260
18854460
5.478225.222685.180038
1
n
mfx i
ii
Cn
bfAx i
ii
8
1
or
42660
)4(2...)2(8)3(35.3078
x
4.314242660
95.3078
Median
Cf
Cfn
LM
2
9.310842614
222
60
5.2865
M
Since the class midpoint of interval which has the largest frequency is 3078.5, mode is 3078.5
Measures of position
What are the 25th, 75th percentiles and the median?
TCLxTCLx
TCLxx
xx
xx
23
2
23
21
True
Class Limit Less than ... Frequency
Less than ... Rel. Freq.
1587.5 0 0.00
2013.5 3 5.00 2439.5 11 18.33 2865.5 22 36.67 3291.5 36 60.00 3717.5 47 78.33 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00
True
Class Limit Less than ... Frequency
Less than ... Rel. Freq.
1587.5 0 0.00
2013.5 3 5.00 2439.5 11 18.33 2865.5 22 36.67 3291.5 36 60.00 3717.5 47 78.33 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00
25=x1
36.67-18.33 25-18.33
x=?
x-2439.52865.5-2439.5
2439.5 18.33
X 25
2865.5 36.67
TCLxTCLx
TCLxx
xx
xx
23
2
23
21
5.24395.2865
5.2439
33.1867.36
33.1825
xX=2594,3
True
Class Limit Less than ... Frequency
Less than ... Rel. Freq.
1587.5 0 0.00
2013.5 3 5.00 2439.5 11 18.33 2865.5 22 36.67 3291.5 36 60.00 3717.5 47 78.33 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00
75=x1x=?
3291.5 60.0
X 75
3717.5 78.33
5.32915.3717
5.3291
0.6033.78
0.6075
x1.3640 x
P75=Q3=3640.1
75% of observations lie below 3640.1.
3291.5 60.0
X 75
3717.5 78.33
If the central %90 were considered as normal birthweight, what are the two limits for normality?
x1 x2
Normal b.w90%
5% 5%
True
Class Limit Less than ... Frequency
Less than ... Rel. Freq.
1587.5 0 0.00
2013.5
3
5.00 4143.5 54 90.00 4569.5 58 96.67 4995.5 60 100.00
.
.
.
.
.
.
.
.
.
x1 x2
Normal b.w90%
5% 5%
x1
95
x2
5.41435.4569
5.4143
0.9067.96
0.9095
xgr 8.4462 x
1
8
1
22
nn
mfmf
s i
iiii
624356925)5.4782(2)5.2226(8)5.1800(3 2228
1
2
i
iimf
63554883994)188544( 2
28
1
i
iimf
73559
6063554883994
624356925
s
Standart deviation
1
)(8
1
22
nn
bfbf
Cs i
iiii
or
177)4(2...)2(8)3(3 2222iibf
81)4(2...)2(8)3(3)( 22 iibf
gr 73559
)60/81(177426
s
Coefficient of variation
%4.231004.3142
735100
x
sCV
The amount variation between observations in this distribution is 23.4% in terms of the mean.
A study was conducted to analyze the relation between coronary heart disease (CHD) and body weight status (obese and normal). Observed values are as follows:
Body weight status
CHD
Total
Yes No
n % n %Normal 4 8 46 92 50
Obese 10 25 30 75 40
Total 14 16 76 84 90
8
25
92
75
01020304050
60708090
100
Normal Obese
Body Weight Status
%
Yes
No
CHD
Distribution of CHD according to body weight status
825
9275
01020304050
60708090
100
Normal Obese
Body Weight Status
%
No
Yes
CHD
Distribution of CHD according to body weight status
Example
In a goiter hyper-endemic area, thyroid gland and its hormones T3, T4 and TSH (Thyroid Stimulating Hormone) were studied in 225 subjects in three groups
• Hypothyroid • Euthyroid• Hyperthyroid
T4 levels of three study groups are observed as follows:
T4 levels of the study groups
Hypothyroid Group 1,1 1,7 2,1 2,7 3,41,1 1,7 2,1 2,7 3,41,1 1,7 2,1 2,8 3,41,1 1,7 2,4 2,9 3,41,1 1,9 2,4 2,9 3,41,2 1,9 2,4 2,9 3,41,2 2,0 2,4 2,9 3,41,4 2,0 2,4 2,9 3,51,5 2,0 2,5 3,0 3,51,5 2,0 2,5 3,1 3,71,5 2,0 2,5 3,1 3,71,5 2,0 2,6 3,1 3,91,5 2,0 2,6 3,4 3,91,6 2,1 2,7 3,4 3,91,6 2,1 2,7 3,4 4,0
Euthyroid Group1,6 4,5 5,5 6,9 8,42,1 4,8 5,6 7,1 8,62,4 4,8 5,7 7,1 8,62,4 4,8 5,7 7,1 8,72,8 4,8 6,0 7,1 9,63,0 4,8 6,1 7,2 9,63,0 4,9 6,1 7,5 9,63,4 5,0 6,1 7,6 9,63,4 5,0 6,4 8,0 9,73,8 5,0 6,4 8,0 9,94,0 5,1 6,4 8,1 9,94,0 5,1 6,6 8,1 10,44,2 5,1 6,7 8,1 10,94,2 5,5 6,7 8,4 11,44,3 5,5 6,9 8,4 12,5
Hyperthyroid Group 9,4 11,2 12,4 13,9 15,09,4 11,2 12,6 13,9 15,09,5 11,2 12,7 14,0 15,09,6 11,3 12,7 14,2 15,1
10,0 11,8 12,7 14,2 15,610,0 11,8 13,0 14,2 15,610,0 11,8 13,2 14,2 16,210,3 12,0 13,2 14,2 16,310,3 12,1 13,4 14,4 16,410,3 12,1 13,5 14,6 16,710,9 12,1 13,5 14,7 16,711,0 12,1 13,5 14,7 16,711,0 12,1 13,5 14,7 16,711,0 12,4 13,6 14,8 17,911,2 12,4 13,6 14,8 18,1
Descriptive Statistics of the study groups for T4
GROUPS Hypothyroid Euthyroid Hyperthyroid Mean 2,46 6,38 13,16 Median 2,40 6,10 13,20 Mode 3,40 4,80 12,1 & 14,2 Std. Deviation 0,82 2,37 2,14 Variance 0,68 5,62 4,58 CV 33,33 37,15 16,26 Minimum 1,10 1,60 9,40 Maximum 4,00 12,50 18,10 Range 2,90 10,90 8,70 Quartiles 1st (25%) 1,70 4,80 11,30 3rd (75%) 3,10 8,10 14,70
Classification of the T4 values in hypothyroid group
Min = 1,1
Max = 4,0
Range = 4,0 – 1,1 = 2,9
# of classes = 8
Class Int. = 2,9/8 = 0,36 0,4
4,
05
3,65
3,25
2,85
2,45
2,05
1,65
1,25
Nu
mb
er
of
Pa
tien
ts
16
14
12
10
8
6
4
2
0
Histogram of T4 values in hypothyroid group
Box plot of T4 values in three groups
HyperthyroidEuthyroidHypothyroid
T4
20
15
10
5
0
Error bars of T4 values in three groups
HyperthyroidEuthyroidHypothyroid
Me
an
+-
1 S
D T
4
18
16
14
12
10
8
6
4
2
0
Hypothyroid Euthyroid Hyperthyroid
T4
20
15
10
5
0
Scatter plot of T4 values in three groups