In statistics, the standard score is the (signed) number of standard deviations an observation or datum is above the mean. Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean. It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization (statistics) for more).Standard scores are also called z-values, z-scores, normal scores, and standardized variables; the use of "Z" is because the normal distribution is also known as the "Z distribution". They are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with μ = 0 and σ = 1), though they can be defined without assumptions of normality.
Understanding the Z- score (Application on evaluating a Learner`s performance)Lawrence Avillano
Application of Z-score or standard score on evaluating a students performance. It includes the formula for z-value, interpretation of in relation to the mean and stdev, sample step by step calculation and interpretationof z-score, and sample real scenario application.
Commonly Used Statistics in Medical Research Part IPat Barlow
This presentation covers a brief introduction to some of the more common statistical analyses we run into while working with medical residents. The point is to make the audience familiar with these statistics rather than calculate them, so it is well-suited for journal clubs or other EBM-related sessions. By the end of this presentation the students should be able to: Define parametric and descriptive statistics
• Compare and contrast three primary classes of parametric statistics: relationships, group differences, and repeated measures with regards to when and why to use each
• Link parametric statistics with their non-parametric equivalents
• Identify the benefits and risks associated with using multivariate statistics
• Match research scenarios with the appropriate parametric statistics
The presentation is accompanied with the following handout: http://slidesha.re/1178weg
In statistics, the standard score is the (signed) number of standard deviations an observation or datum is above the mean. Thus, a positive standard score represents a datum above the mean, while a negative standard score represents a datum below the mean. It is a dimensionless quantity obtained by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing (however, "normalizing" can refer to many types of ratios; see normalization (statistics) for more).Standard scores are also called z-values, z-scores, normal scores, and standardized variables; the use of "Z" is because the normal distribution is also known as the "Z distribution". They are most frequently used to compare a sample to a standard normal deviate (standard normal distribution, with μ = 0 and σ = 1), though they can be defined without assumptions of normality.
Understanding the Z- score (Application on evaluating a Learner`s performance)Lawrence Avillano
Application of Z-score or standard score on evaluating a students performance. It includes the formula for z-value, interpretation of in relation to the mean and stdev, sample step by step calculation and interpretationof z-score, and sample real scenario application.
Commonly Used Statistics in Medical Research Part IPat Barlow
This presentation covers a brief introduction to some of the more common statistical analyses we run into while working with medical residents. The point is to make the audience familiar with these statistics rather than calculate them, so it is well-suited for journal clubs or other EBM-related sessions. By the end of this presentation the students should be able to: Define parametric and descriptive statistics
• Compare and contrast three primary classes of parametric statistics: relationships, group differences, and repeated measures with regards to when and why to use each
• Link parametric statistics with their non-parametric equivalents
• Identify the benefits and risks associated with using multivariate statistics
• Match research scenarios with the appropriate parametric statistics
The presentation is accompanied with the following handout: http://slidesha.re/1178weg
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
Please Subscribe to this Channel for more solutions and lectures
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.3: Estimating a Population Standard Deviation or Variance
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
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students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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2. INTRODUCTION
A Z-score is a numerical measurement that
describes a value's relationship to the mean of a
group of values.
Z-score is measured in terms of standard deviations
from the mean.
If a Z-score is 0, it indicates that the data point's
score is identical to the mean score.
3. How
to
calculate
the
z-score? A Z-score tells where the generic normal
distribution curve is present.
Most Z-scores are between -3 and +3
(because 99.7% of the data is between -3
and +3).
4. FORMULA
OF
Z-SCORE
For Two - Sample Where,
x
̄ 1 = Mean of the First Sample
x
̄ 2 = Mean of the Second Sample
μ1 = Mean of the First
Population
μ2 = Mean of the Second
Population
(μ1 – μ2) = Hypothesized
Difference Between the
Population Means
σ1 = Standard Deviation of the
First Population
σ2 = Standard Deviation of the
Second Population
For Standard Error of the Mean
Where,
“x” = Raw Score,
“µ” = Population Mean,
“σ” = Standard Deviation
“n” = Number of Observations
For One - Sample
z =
x - µ
σ
Where,
“x” =Raw Score,
“µ”=Population Mean,
“σ”=Standard Deviation
Formula
of
Z-Score
8. How
to
interpret
Z
–
Score
in
Python?
How
to
interpret
Z-Score
in
Python?
Data : [1, 8, 9, 7, 6, 5, 4, 3, 7, 8]
Mean: 5.8
Standard Deviation : 2.5298221281347035
Zscore : [-2. 0.91666667 1.33333333 0.5 0.08333333
-0.33333333
-0.75 -1.16666667 0.5 0.91666667]
RUN
import statistics
import scipy.stats as stats
A = [1,8,9,7,6,5,4,3,7,8]
print ("Data :",A)
mean = statistics.mean(A)
print ("nMean:", mean)
stdev = statistics.stdev(A)
print ("nStandard Deviation :",stdev)
zscore = stats.zscore(A)
print ("nZscore :", zscore)
9. How
to
interpret
Z-Score
in
R
Studio?
a <- c(9, 10, 12, 14, 5, 8, 9)
mean(a)
sd(a)
a.z_score <- (a - mean(a)) / sd(a)
plot(a.z_score, type="o", col="green")
> a <- c(9, 10, 12, 14, 5, 8, 9)
> mean(a)
[1] 9.571429
> sd(a)
[1] 2.878492
> a.z_score <- (a - mean(a)) / sd(a)
> plot(a.z_score, type="o", col="green")
RUN
10. Why
is
z-score
important?
It is useful to standardize the values (raw scores) of a normal distribution by
converting them into z-scores because:
Importance
11. Example
1
Suppose SAT scores among college students are normally distributed
with a mean of 500 and a standard deviation of 100.If a student scores
a 700,what would be her z-score?
Z-score = Where,
x = 700, µ = 500, σ = 100
=2
Her z-score would be 2 which means her score is two standard
deviations above the mean.
700 - 500
100
Example
1
12. Example
2
A set of Math test scores has a mean of 70 and a standard deviation of 8.
And a set of English test scores has a mean of 74 and a standard deviation of 16.
For which test would a score of 78 have a higher standing?
Find the z-score for each test:
z-score of Math Test= =1
z-score of English Test= =0.25
The Math score would have the highest standing since it is 1
standard deviation above the mean while the English score is
only 0.25 standard deviation above the mean.
78-70
8
78 - 74
16
Example
2
13. Example
3
A company wanted to compare the performance of its call center employees in two
different centers located in two different parts of the country – Hyderabad, and
Bengaluru, in terms of the number of tickets resolved in a day (hypothetically
speaking). The company randomly selected 30 employees from the call center in
Hyderabad and 30 employees from the call center in Bengaluru. Calculate the P-Value.
Hyderabad: x
̄ 1 = 750, σ1 = 20
Bengaluru: x
̄ 2 = 780, σ2 = 25
First, we will formulate the null and alternate hypotheses and set the level of significance for the test.
H0: There is no difference between the performance of employees at different call centers.
H1: There is a difference in the performance of the employees.
The level of significance is set as 0.05.
Next, the mean and standard deviation for each sample will need to be determined.
Hyderabad: x
̄ 1 = 750, σ1 = 20
Bengaluru: x
̄ 2 = 780, σ2 = 25
Next, we will calculate the hypothesized difference between the two population means.
In this case, the company is hypothesizing that the mean performance in Hyderabad is the same as that of
Bengaluru. So, (μ1 – μ2 ) = 0
Finally, we will use the formula for two-sample z-test for means to calculate the test statistic.
z= (x
̄ 1 – x
̄ 2 ) / √((σ1 )²/n1 + (σ2)²/n2)
z = (-30) / √((20)²/30 + (25)²/30))
z = -5.13
At a significance level of 0.05, the p-value is less than 0.00001. As the p-value is lot less than the critical
value of 0.05, the result is statistically significant and hence you can reject the null hypothesis. Hence,
the performance of Hyderabad’s team is considered to be not equal to the performance of Bengaluru’s
team.
14. Example
3
In general, the mean height of women is 65″ with a standard deviation of
3.5″. What is the probability of finding a random sample of 50 women with
a mean height of 70″, assuming the heights are normally distributed?
z = (x – μ) / (σ / √n)
= (70 – 65) / (3.5/√50) = 5 / 0.495 = 10.1
The key here is that we’re dealing with a sampling distribution of
means, so we know we have to include the standard error in the
formula. We also know that 99% of values fall within 3 standard
deviations from the mean in a normal probability distribution
(see 68 95 99.7 rule). Therefore, there’s less than 1% probability
that any sample of women will have a mean height of 70″.