Intro to Z-scores

Summary

This video introduces the concept of Z-scores, focusing on their importance in statistics, particularly with normal, quantitative data. It explains how Z-scores are calculated using the mean and standard deviation, and how they help compare individual data points to the group average. The presenter uses examples like IQ scores to demonstrate Z-score calculation and interpretation, emphasizing how Z-scores indicate whether a value is above or below the mean, and how they help identify typical or unusual data points. The video provides a foundational understanding for further discussions on Z-scores.

Takeaways

• 📊 Z-scores are crucial for normal quantitative data and are used in various statistical scenarios like critical values, confidence intervals, and test statistics for proportions.
• 📉 Z-scores are based on normal distribution, where the mean and standard deviation accurately represent the data. The data must be normally distributed for the Z-score to be valid.
• 📏 The formula for calculating a Z-score involves taking the data value, subtracting the mean, and dividing by the standard deviation.
• 👍 A positive Z-score indicates a value above the mean, while a negative Z-score means the value is below the mean.
• 🧠 Example: In an IQ test with a mean of 100 and a standard deviation of 15, Maria’s IQ of 147 has a Z-score of 3.13, meaning her score is 3.13 standard deviations above the mean.
• 📐 Z-scores can be used to identify outliers, where values two standard deviations above or below the mean (Z-scores greater than or equal to 2 or less than or equal to -2) are considered unusual.
• 👥 Z-scores between -1 and 1 are typical, as they represent data that falls within one standard deviation of the mean, covering around 68% of normally distributed data.
• 🧮 Z-scores are not percentages, proportions, or units like dollars or miles. They are measured in terms of standard deviations, a way to standardize and compare data across different scales.
• 🔍 Z-scores also help in determining statistical significance, where Z-scores beyond certain thresholds indicate significantly high or low data points.
• 📚 The script emphasizes the importance of understanding Z-scores as a tool for comparing data and identifying whether data points are typical, unusual, or outliers.

Q & A

What is a z-score?

A z-score represents the number of standard deviations a data point is from the mean. It is used to compare an individual data point to the overall dataset.

When should z-scores be used?

Z-scores should be used when working with normal or bell-shaped data, as the calculation relies on accurate mean and standard deviation values.

How do you calculate a z-score?

To calculate a z-score, subtract the mean from the data value, then divide the result by the standard deviation. The formula is: (data value - mean) / standard deviation.

What does a positive z-score indicate?

A positive z-score indicates that the data value is above the mean.

What does a negative z-score indicate?

A negative z-score indicates that the data value is below the mean.

How do you interpret z-scores in terms of outliers?

A z-score greater than or equal to 2 indicates a high outlier, while a z-score less than or equal to -2 indicates a low outlier.

What is considered a typical z-score range?

A typical z-score falls between -1 and 1, which corresponds to the middle 68% of values in a normal distribution.

How would you calculate the z-score for Maria's IQ of 147 if the mean is 100 and the standard deviation is 15?

To calculate Maria's z-score, subtract 100 from 147 to get 47, then divide 47 by 15. The result is a z-score of 3.13, indicating that Maria's IQ is 3.13 standard deviations above the mean.

What does a z-score of 3.13 for Maria’s IQ mean?

Maria’s z-score of 3.13 means her IQ is 3.13 standard deviations above the mean, indicating that she has an unusually high IQ compared to the general population.

How do z-scores relate to significance in statistics?

Z-scores are often used in significance testing. Values greater than or equal to 2 (or less than or equal to -2) are considered unusual and may indicate statistical significance.

Outlines

00:00

📊 Introduction to Z-Scores and Their Importance in Statistics

This paragraph introduces the concept of Z-scores, highlighting their significance in statistics, especially in analyzing normal quantitative data. The Z-score is presented as a critical value and test statistic, particularly for proportions, and is useful for comparing individual data points to the overall distribution. It emphasizes the importance of having normally distributed data for the Z-score to be accurate, as the calculation is based on the mean and standard deviation of the data.

05:01

🧮 Calculating Z-Scores with an Example

This paragraph explains how to calculate a Z-score using a practical example involving IQ scores. Maria's IQ of 147 is compared to the mean IQ of 100 with a standard deviation of 15, resulting in a Z-score of 3.13. The paragraph also touches on the rounding conventions in Z-score calculations, influenced by historical practices. The positive Z-score indicates that Maria's IQ is above the average, illustrating the concept of standard deviations as a measure of comparison.

10:03

📈 Interpreting Z-Scores and Their Significance

Here, the paragraph discusses the interpretation of Z-scores, specifically what a positive or negative Z-score indicates. It further elaborates on how Z-scores can help identify outliers and unusual values in data. The concept of typical values, represented by Z-scores between -1 and 1, is introduced. The paragraph uses Maria's high Z-score to illustrate how values above 2 are considered unusually high, thus making Maria's IQ unusually high.

🧠 Z-Scores and Typical Versus Unusual Values

The final paragraph provides another example using Rick's IQ of 87, which yields a negative Z-score of -0.87. This score is within the typical range (-1 to 1), meaning Rick's IQ is typical compared to the general population. The paragraph emphasizes the importance of understanding that Z-scores represent standard deviations, not percentages or proportions, and clarifies that not all data points fall into typical or unusual categories. This serves as a conclusion to the introductory discussion on Z-scores, with a promise of deeper exploration in future lessons.

Keywords

💡Z-score

A Z-score measures how many standard deviations a data point is from the mean of a data set. It is used to standardize scores in a normal distribution, allowing comparison of individual values across different data sets. In the video, Z-scores help determine how an individual's score, like IQ, compares to the average population.

💡Standard deviation

Standard deviation measures the amount of variation or dispersion in a set of values. A low standard deviation means the data points are close to the mean, while a high standard deviation indicates that they are spread out. In the video, it is crucial for calculating Z-scores, as the Z-score is the number of standard deviations a data value is away from the mean.

💡Mean

The mean is the average of a data set, calculated by summing all values and dividing by the number of values. In the video, the mean is used as a reference point for calculating Z-scores, which measure how far a particular data point is from the average value.

💡Normal distribution

A normal distribution is a bell-shaped curve where most data points cluster around the mean, with fewer data points appearing as you move away from the center. Z-scores are specifically used with normally distributed data, as mentioned in the video, to assess how typical or unusual a particular data value is.

💡Critical values

Critical values are the cutoff points in hypothesis testing that determine whether a test statistic falls in the rejection region. In the video, critical values are linked to Z-scores and confidence intervals, helping identify whether a result is statistically significant.

💡Confidence intervals

A confidence interval is a range of values within which a population parameter is expected to lie, with a certain degree of confidence. Z-scores are used to calculate confidence intervals in statistics, especially in determining whether a population parameter falls within a given range.

💡Population mean (μ)

The population mean, denoted by the Greek letter μ (mu), is the average of all values in a population. It is contrasted with the sample mean (X̄) in the video, where Z-scores are sometimes calculated using the population mean instead of the sample mean to understand how individual data points compare to the overall population.

💡Sample standard deviation (s)

The sample standard deviation, denoted by 's', is a measure of how spread out sample data is from the sample mean. In the video, the sample standard deviation is essential for Z-score calculations when working with sample data, as it helps measure the relative position of individual data points within the sample.

💡Unusual values

Unusual values are data points that lie two or more standard deviations away from the mean. These values are often considered outliers or significant in the context of statistical analysis. In the video, Maria’s IQ is considered unusually high because her Z-score is greater than 2, making her score an outlier.

💡Typical values

Typical values are data points that fall within one standard deviation of the mean. These values are considered common or expected within a normal distribution. In the video, Rick’s IQ score has a Z-score of -0.87, which places him within the typical range, indicating his IQ is close to the average.

Highlights

Introduction to Z-scores and their widespread use in statistics for normal quantitative data, critical values, confidence intervals, and test statistics.

Z-scores require data to be normally distributed for accuracy, as they rely on the mean and standard deviation.

Mean and standard deviation are the most accurate descriptors of normal data distributions.

The formula for a Z-score is the data value minus the mean, divided by the standard deviation.

Positive Z-scores indicate a value above the mean, while negative Z-scores indicate a value below the mean.

Example of calculating a Z-score: Maria's IQ of 147, with a mean of 100 and standard deviation of 15, results in a Z-score of 3.13.

The importance of rounding Z-scores to the hundredths place, often a tradition from the pre-computer era when Z-score charts were used.

A Z-score represents the number of standard deviations a value is from the mean, making it a standardized measure.

Z-scores can be used to identify outliers: values more than 2 standard deviations from the mean are considered unusual.

Typical values in a normal distribution have Z-scores between -1 and 1, encompassing about 68% of the data.

Maria's Z-score of 3.13 indicates an unusually high IQ, as it exceeds the threshold of 2 standard deviations above the mean.

Rick's IQ of 87 results in a Z-score of -0.87, indicating a typical IQ, as it falls between -1 and 1.

A Z-score of 1.5 is neither typical nor unusual, falling in a gray area between typical and outlier values.

Z-scores provide a standardized comparison, allowing values from different data sets to be compared on the same scale.

Z-scores will be used throughout the course, both for calculation and interpretation in various statistical contexts.

Transcripts

hi everyone this is Matty show with

intro stats today we are looking at Z

scores Z scores they're very famous

especially for normal quantitative data

we also use them in a lot of situations

critical values and confidence intervals

we also use them as test statistics

mainly for proportions so there's a lot

of uses of Z scores in statistics and so

I wanted to kind of introduce what is

the z-score and sort of how does it work

so the first thing to remember is

z-scores really go with normal data the

data does have to be normal for the

z-score to be accurate if you remember

last time when we talked about normal

quantitative data we said that the most

accurate average receptor is the mean

and the most accurate spread is the

standard deviation and those two

statistics were only accurate if the

data was normal or bell-shaped the the

z-score calculation is based on the mean

and standard deviation being accurate so

you want to make sure that your data is

normal before you start looking at

z-scores all right so just a couple

things we saw the last time that the

mean of a dataset is often denoted as an

X with a bar over it that usually that

symbol just means the mean of a sample

data set also we saw that the standard

deviation is s or the sample standard

deviation occasionally though you will

see and we'll kind of get more into this

in the next next unit but there are

other letters that you'll see in stats

this letter right here that looks like a

U of the tail is the Greek letter mu and

it's often denoted as a population mean

so if you knew the population mean

average this symbol here is another

Greek letter called Sigma again we'll

get more into these letters in the next

unit so don't worry too much about it

right now but this this this letter

right here this Greek letter Sigma

is usually denoted as a population

standard deviation so if you're talking

about standard deviation of the entire

population and that would be Sigma so

you'll see these letters sometimes in

z-score formulas in stat books all right

so what's a z-score so a z-score

basically counts the number of standard

deviations above or below the mean okay

so it's really used as sort of a

comparison number if you want to see how

you did compared to everybody else a

z-score is one way to go so you take

basically it's calculated by taking the

data value like you're you know if you

if you ran a marathon right you wanted

to see how did I do compared to

everybody else well you could take your

time in the marathon minus the mean

average of the all the times in the

marathon divided by the standard

deviation of all the times in the

marathon and you'd get a z-score and

that z-score would be able to tell you

how you did compared to everybody else

that's kind of what we use this board

now the data value is sometimes denoted

as an X and I think if you remember when

we were calculating mean and standard

deviation I was using the letter X X

minus the mean this so X is like your

marathon time right the data value your

that you're come that you're looking at

and then the mean is X bar and s is

standard deviation also sometimes you'll

see the formula in stat books as X minus

mu whenever that's the population mean

divided by Sigma the population standard

deviation to me though especially for

intro students I would go with the word

the word formula right that one you

never get messed up so you know you

remember that you know and I'm

remembering all the letters yet but just

name is the data value minus the mean

divided by the standard deviation okay

now if this z-score actually comes out

positive the data value must have been

above the mean and if the if the z-score

comes out negative

then the data value must be below the

mean so kind of keep in mind

that and when you're kind of explaining

the z-score the positive z-score you're

going to say above in the sentence and

negative z-scores are going to say below

in the sentence

okay so just kind of keep that in mind

so let's look at a quick couple quick

examples so let's suppose we're going to

look at IQ tests are normally

distributed or normal with a mean of 100

and a standard deviation of 15 right so

a mean of 100 and the standard deviation

of 15 so so let's suppose that we look

at Maria's IQ and Maria's IQ came out to

be 147 how is that how does that compare

to everybody else that takes an IQ test

well we could calculate the z-score for

Maria so all you do is you put in the

Maria's score 147 minus the mean so

minus 100 and then divide by 15 right

divided by 15 so if we did that 147

minus 100 is is 47 and 47 divided by 15

you get positive 3 point 1 3 3 3 3 3 3 3

and that's the z-score if you notice you

didn't really have to put this little

positive sign I do that anytime a lot of

times in certain statistics the positive

and negative is really really important

in terms of interpretation so a lot of

times I will make sure to put a little

symbol next to it just reminding myself

that it's a positive value or it's a

negative value that's really important

with z-scores now if you'll notice I did

round it I wryly round it to the

hundreds place the second number to the

right of the decimal not for really any

good reason in the old days before

computers we used to have these charts

that you would look up things and that

were organized by z-score and you would

look up the z-score on these charts and

the charts were always rounded to the

hundreds place so if you're like me and

you've been around for a while have you

been kind of doing stats for a while you

may be

looking up stuff you may have looked up

stuff on those charts and those charts

were always rounded to the hundreds

place so I think a lot of us old-timers

that were been doing this before

computers we not that I actually was

doing charts before computers because

computers have been invented but I have

used the charts in the past and then

again I those charts were grounded to

the hundreds place so that's why I

rounded it to hundreds plus now the more

important part of this is what does this

mean right what does this mean first of

all notice the z-score was positive that

means Maria's IQ was above the mean

right she was above the average so if we

look at that okay well positive means

it's above right and but remember a

z-score is not a percentage a z-score is

not kilograms it's not dollars it's not

miles a z-score is number of standard

deviations that's why we often call it a

standardizing score it's a way of

comparing things when you may not

understand like you might not understand

the physics involved in in some data

that maybe you got but maybe if but if

you understand that in terms of number

of standard deviations then you can

still kind of make a night get an idea

of what's going on so Maria's IQ is

three point one three standard

deviations above the mean now that would

be how I would explain it notice I use

the word above because my z-score was

positive and obviously Maria's IQ was

above the mean okay

now you can use z-scores to figure out

outliers and unusual values if you got

if you guys remember when we did our

mean average and standard deviation

video and normal data we said that

anything that's two standard deviations

away from the mean or above is

considered a high outlier and anything

that's two standard deviations below the

mean or or less would be considered a

low outlier so if you translate that

into a z-score that means your z-score

would have to be greater than or equal

to two for it to be unusually high and

then z-scores are less than or equal to

negative two when they're unusually low

now later we'll see that you this two

and negative two we can get a little

more accurate with those later on we'll

get into critical values and things like

that but right now just having your head

okay about two or more standard

deviations away is unusual it's also

considered significant so we'll kind of

get into that later two z-scores are

sometimes used for significance measures

if you guys remember the typical values

in a normal data are between our one

standard deviation from the mean

so that would be translated that into a

z-score typical values would have a

z-score between negative 1 and positive

1 so let's go back to Moorea Moorea

z-score was 3 point 1 3 which is

definitely higher than 2 right so that

means she was unusually high Maria has

an unusually high IQ okay because her Z

score was above 2 so that makes sense

like you might not understand like when

I looked at 147 I didn't know is that a

lot or is that not a lot now I know it's

a lot right because the z-score tells me

let's look at another one so Rick's IQ

is 87 what would be Rick's z-score

well again you start with Rick's value

right Rick's value is 87 you minus the

mean and then you divide by the standard

deviation so 87 minus 100 is negative

13/15 we get negative zero point eight

six six six six six six

again I rounded it to the hundredths

place so I got negative zero point eight

seven

now be careful this is not a proportion

this is not a percentage do not convert

that to 87 percent a z-score is number

of standard deviations it's not actually

a

or a proportion you want to be very

careful with that you leave it as

negative 0.87 that means that Rick's IQ

is point eight seven standard deviations

below the mean below the mean

notice again negative means below notice

I didn't say negative 0.87 standard

deviations below the negative tells me

it's below the point eight seven tells

me how many standard deviations so

better to say that it's point eight

seven standard deviations below the mean

now where is Rick fall compared to other

people well didn't we say any z-score

between negative one and positive one

would be considered typical and this IQ

negative point eight seven is between

negative one and positive one on the

number line so Rick is actually very

typical he has a typical IQ like a lot

of people but I think we mentioned in

the normal data section though that

that's about the middle 68% so Rick's

kind of in the middle 68% of people

people's IQ okay now remember not we

talked about this not everybody is

unusual or typical there's people that

are sort of in that middle ground so

it's like suppose I have a z-score of

1.5 well 1.5 is not typical it's it's

not in the typical zone but it's also

not unusual right because not two or

above so a 1.5 z-score is not typical

and it's not unusual right so don't

think that everybody has to fall into

typical or unusual okay all right so I

hope this helped you as the z-scores

we'll be talking more about z-scores

throughout the class but this is just an

introduction to calculating them in an

introduction just sort of starting to

explain them like I say will get more

and more into z-scores throughout the

class all right