Standard Error in Research its calculation and inference

This Error discusses on Standard Error in Research. Its Calculation and Uses. Standard Error of Mean, Standard Error of Median, Standard Error of Correlation etc.

1. Standard Error
in Statistical Calculation:
Understanding Data and
Making Inferences
by Dr. Vikramjit Singh

2. The Essence of Standard Error
•Statistical inference allows us to generalize from a sample to a larger population. For example,
we can use data from a group of students to estimate the performance of all students in a
particular school district.

•Measures calculated from the sample is called as Statistics- “fluctuations of sampling”

•Measures descriptive of a population , are called as parameters – “ fixed reference values”

•“ How good we are to estimate the parameters based on the sample ?”


• Relevance of calculating standard error and confidence intervals – to ascertain the
trustworthiness of the estimates

3. The Standard error of the mean (??????
� )
The mean is the crucial measures of central tendency.
The mean represents the average value in a dataset.
SE
M or ??????� =
??????
ℕ


σ=standard deviation of the population
N = Number of cases of the Sample
As population SD is not available
we use the sample SD which a
representation of Population SD

To correct the underestimation of
SD of population specifically for a
small Sample we calculate SD of
the sample by the formula
�=
??????
2
??????−1
in place of �=
??????
2
??????

4. Ex- The mean of a test of abstract reasoning for 225
boys in the tenth grade of city F was 27.26 with a SD
of 11.20 . How dependable is this mean? Specifically,
how good an estimate is it of the mean which could
be expected if all the tenth – grade boys in city F
were tested ?
Source : Garrett, H.E.
??????
� =
??????
ℕ
, ??????
� =
11.20
225
,

??????
� = 0.75

5. Sample Mean = 27.26
??????
?????? =
??????
ℕ
, ??????
?????? =
��.��
���
, ??????
?????? = 0.75


Setting Up confidence interval for the
Population Mean
Proportion of cases from the Mean and
deviation from the Mean in the NPC in
one side of the curve
95 % cases lie at ±1.96
???????????? �????????????� distance
Confidence interval for the Population Mean

Population Mean = Sample Mean ±
1.96 x ????????????
= 27.26 ± 1.96 x 0.75
= 27.26 ± 1.47
28.73 or 25.79

Population Mean lie between 28.73
and 25.79
with 95 % probability

6. Sample Mean = 27.26
??????
?????? =
??????
ℕ
, ??????
?????? =
��.��
���
, ??????
?????? = 0.75


Setting Up confidence interval (fiduciary limit or
fiduciary probability )for the Population Mean
Proportion of cases
from the Mean and
deviation from the
Mean in the NPC in
one side of the curve
99 % cases lie at ±�.�??????
???????????? �????????????� distance
Confidence interval for the Population Mean

Population Mean = Sample Mean ±
2.58 x ????????????
= 27.26 ± 2.58 x 0.75
= 27.26 ± 1.935
29.195 or 25.325

Population Mean lie between 29.195
and 25.325
with 99 % probability

7. The Standard error of the Median (??????
�???????????? )
The median represents the middle value in a dataset.
SE
Mdn or ??????
�???????????? =
1.253??????
ℕ
=
1.858�
ℕ



σ=standard deviation of the population
Q= Interquartile Range
N = Number of cases of the Sample

8. Applying SE
Mdn in Large Samples and getting the confidence interval
Ex- On the Trabue language Scale A, 801 twelve year old boys made
the following record: Median =21.40 and Q =4.90. how well does the
median represent the median of the population from which the
sample was drawn.
Source : Garrett, H.E.
Median = 21.40 , Q = 4.90 and N =
801
??????
�???????????? =
1.858�
ℕ
=
1.858 ?????? 4.90
801
= 0.32

Confidence interval for the
Population Median
Population Median = Sample Median ± 2.58 x ??????
??????????????????
=21.40 ± 2.58 x 0.32
= 21.40 ± 0.825
22.225 or 20.575

Population Mean lie between 22.225 and 20.575
with 99 % probability
Population Median = Sample Median ± 1.96 x
??????
??????????????????
=21.40 ± 1.96 x 0.32
= 21.40 ± 0.627
22.027 or 20.773

Population Median lie between 22.027 and
20.773
with 95 % probability

9. The Standard error of the Standard Deviation (SE
?????? /??????
??????)
??????
?????? =
0.71??????
??????

σ=standard deviation
N = Number of cases of the Sample
Ex from the previous problem - The mean of a
test of abstract reasoning for 225 boys
in the tenth grade of city F was 27.26
with a SD of 11.20 . How dependable is
this mean? Specifically, how good an
estimate is it of the mean which could
be expected if all the tenth – grade boys
in city F were tested ?
Source : Garrett, H.E.
??????
?????? =
0.71??????
??????
= =
0.71 ?????? 11.20
225
= 0.53
SD
pop confidence interval with 99%
probability

SD
pop = SD
Sample ± 2.58 x 0.53
= 11.20 ± 1.367
= 12.567 or 9.833

Population SD lie between 12.567 and 9.833
with 99 % probability

10. The Standard Error of Quartile Deviation
??????
?????? =
0.786??????
??????
=
1.17�
??????

σ=standard deviation
N = Number of cases of the Sample
Q= Interquartile Range
Ex from the previous problem -
On the Trabue language
Scale A, 801 twelve year
old boys made the
following record: Median
=21.40 and Q =4.90. how
well does the median
represent the median of
the population from which
the sample was drawn.
Source : Garrett, H.E.
??????
?????? =
1.17�
??????
=
1.17 ?????? 4.90
801
= 0.203


With 99% probability the
Population Q lies in between
4.38 to 5.42 i.e.

4.90 ± 2.58 x 0.203

11. The Standard Error of Percentage
??????
% =
��
??????
=
41.4% ?????? 58.6 %
348
=2.6 %
??????
% =
��
??????

P=Percentage of Occurrence
of the Behaviour
Q = (1-P)
N= Number of cases of the
Sample
Ex- In a study of cheating among students , 144 or 41.4% of the 348
children from homes of high socio-economic status were found to
have cheated on various tests. Assuming our samples to be
representative of children from good homes, how much confidence
can we place in this percentage? How well does it represent the
population percentage?
Source : Garrett, H.E.
0.99 confidence interval of the population percentage is
41.4% ±�.�?????? ?????? �.�% i.e. 34.7 % to 48.1%
So with 99% surety the children who cheat in this
exam ranges from at least 34.7 % and not larger
than 48.1%

12. The Standard Error of the
Coefficient of Correlation
When r is closer to 0 or less value
with large N –

??????
�=
1−�
2
�


Ex= Suppose the correlation
between height and weight is 0.60
with a sample of 120


??????
�=
1−0.60
2
120
= 0.06

So the 0.99 confidence interval for
the population r can be taken as r
± 2.58 ??????
�

i.e., 0.60 ± 2.58 x 0.06 , range ifs
from 0.45 to 0.75
When r is closer to 1 or -1 , i.e .if it is too high or too
low and N is also low–

Use of fisher’s z function

??????
??????=
1
�−3


Ex= Suppose the correlation between height and
weight is 0.85 with a sample of 52

??????
??????=
1
52−3
= 0.14
From the table of Fisher’s z function - Fisher’s z
value to corresponding value for the r value of 0.85
is 1.26
So the 0.95confidence interval for the population z
can be taken as z ± 2.58 ??????
?????? = 1.26 ± 2.58 x 0.14
= 0.99 to 1.53
Converting these values to r’s from the table
i range ifs from 0.76 to 0.91

13. Thank You


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