Derived
Score Z Scores
T score Standing Score
Derived scores like Z score, T score and standing
score are statistical measures used to standardize and compare data. z score is the
standardization from the population raw data or more than 30 sample data to a
standard score, while the t score is the standardization from the sample data
of less than 30 data to a standard score. z score ranges from -3 to 3, while
the score ranges from 20 to 80.
Z score
A
z score measures how many standard deviations a data point is from the mean of
a group of data. It helps to understand where a particular data point stands
within a distribution.
If
raw score = mean Z score is zero
If
raw score > mean Z score is positive
If
raw score < mean Z score is negative.
Advantages
of Z score
· standardization
· comparison and ranking
· used for normal distribution
· probability calculations
· threshold identification
· comparing different scales
· data transformation
Limitations
of Z score
· assumption of normal distribution
· sensitive to outlier
· dependence on sample size
· assumption of interval or ratio scale
· interpretation complexity
· not robust against skewed data
· dependency on parameters
Types
of Z score table
· positive z score
The observed value is above
the mean of total values.
· negative z score
The observed value is below
the mean of total values.
T score
T-scores are similar to z-score but are used when
the sample size is small. They represent the number of standard deviations a
data point is from the mean, considering the smaller sample size. It indicates
how many SD units an examines score is above or below the mean
Advantages of t small size sample
·
flexibility
in hypothesis testing
·
accommodates unknown parameters
·
reduced
sensitivity to outlier
·
wider
applicability
·
accurate
in non normal distributions
·
adjustments
for degrees of freedom
·
commonly
used in research
·
scores
Disadvantages of t score
·
sensitive
for assumptions
·
impact
of outlier
·
requires
random sampling
·
dependence
of equal variances
·
cautious
interpretation with small sample size
·
type 1
error risk with multiple testing
·
assume
interval or ratio data
·
limited
to comparing means
·
assume
independence
Types of t score
·
one
sample t test
This
test compare data to a theoritical mean
·
two sample t test
This compare the means of two group of data
·
paired t
test
This test compare the mean of the same data group changes
Standard score
This is a generic term for score that have been
standardized to have a mean of 0 and a standard deviation of 1 .both z score
and t score are types of standard scores.
Formula for calculating z score t score
Z score
z = (x - /mu}{/sigma)
z = Z score
x= individual data point
/mu mean of the distribution
/sigma standard deviation of the distribution
This formula provided a standardize measure of how
far a data point is from the mean in terms of standard deviation.
T scores
t = (x -/bar {x} {s/{n})
t =t score
X sample mean
/bar {x} population mean
s sample standard deviation
N sample size
This formula assesses how far the sample mean
deviates from the population mean in term of standard errors
Z score and t score are both used in hypothesis
testing
Key Differences between Z score and
t score
|
Z| score |
T score |
|
Standardization from
population data |
Standardization from
sample data |
|
Population is known |
Population is not
known |
|
Average is always
zero |
Average is always 50 |
|
It ranges from -3 to
3 |
It ranges from 20 and
80 |
|
Standard deviation is
always 1 |
Standard deviation is
always 10 |
|
The derived result
can be negative |
The derived result
can never be negative |
|
|
|
