The Importance of Understanding Scales of Measurement in Statistics

It is crucial to know what type of variable you are dealing with and emphasize the importance of looking at your distributions in your variables. Additionally, it is equally important to keep in mind what scale of measurement you are dealing with. For instance, I presented histograms for temperature measurements in both degrees Fahrenheit and Celsius, highlighting that these are two different scales of measurement.

Scales of Measurement in Statistics

In statistics, we often deal with various scales of measurement. It's nice to know that there is a standard scale, which is called the z-scale. Any score from any scale can be converted into a z-score using z-scores. This allows for efficient communication across statisticians and scientists. By converting all scores to a common metric, it becomes very easy to interpret and simple to calculate.

Converting Raw Scores to Z-Scores

To convert or call a raw score to a z-score, we simply take the raw score, subtract the mean, and divide by the standard deviation. The formula for this conversion is straightforward: we take the raw score, subtract the mean, and then divide by the standard deviation. This process allows us to put any distribution on a common metric, known as the z-scale.

Properties of Z-Scores

One of the nicest aspects of z-scores is that the mean z-score in any one sample is always going to be zero. When we take a raw score and subtract its mean, if the result is zero, it means that the raw score is equal to the mean. For example, if we consider body temperature as our variable, with an average of 98.6 degrees Fahrenheit, taking the raw score (body temperature) and subtracting the mean would give us a result of zero, since 98.6 minus 98.6 equals zero.

Interpreting Z-Scores

The z-score also tells us if a score is above or below average. If we have a negative z-score, it means our score is below average, while a positive z-score indicates that our score is above average. This interpretation makes it easy to understand and communicate data among statisticians and scientists.

Converting Data from Different Scales

To demonstrate the power of z-scores, let's assume we have a normal distribution of healthy individuals with an average body temperature of 98.6 degrees Fahrenheit and a standard deviation of about half a degree (0.5). We then take one individual at random from this distribution whose body temperature is 99.6 degrees. To convert this raw score into a z-score, we apply the formula: take the raw score, subtract the mean, and divide by the standard deviation.

Converting Raw Scores to Z-Scores Example

To convert our raw score of 99.6 degrees Fahrenheit to a z-score, we follow the same steps outlined earlier. We first take the raw score (99.6), then subtract its mean (98.6), resulting in 1 degree difference. Finally, we divide this result by the standard deviation (0.5) to obtain our z-score: 1 / 0.5 = 2. This means that the individual's body temperature of 99.6 degrees Fahrenheit is two units above the average.

Direct Conversion from Different Scales

It's essential to note that when converting data from different scales, we get the same exact numbers. For instance, if we take our previous example and convert it from Fahrenheit to Celsius, with an average body temperature of 98.6 degrees Fahrenheit, we would still obtain a z-score of two for a raw score of 99.6 degrees Fahrenheit. This demonstrates the direct conversion and consistency between different scales when using z-scores.

The Importance of Average in Z-Scores

Lastly, it's crucial to emphasize that the average is zero in any z-score distribution. This means that if we have a normal distribution, with its mean as the central value, subtracting this mean from our raw scores will always result in zero. This property highlights the significance of understanding and using z-scores effectively in statistics.

In conclusion, z-scores offer an effective way to communicate and analyze data across different scales of measurement. By grasping their properties and using them correctly, we can efficiently compare and interpret data among various statistical distributions.

"WEBVTTKind: captionsLanguage: eni've sort of been emphasizing how important it is to know what type of variable you're dealing with and emphasizing how important it is to look at your distributions in your variables it's also really important to keep in mind what scale of measurement you're dealing with so for all those histograms i presented the histograms first in degrees fahrenheit then in degrees celsius those are just two different scales of measurement to measure temperature right so i think we're all used to dealing with scales of measurement but in statistics what's nice is there's a standard scale and it's called the z scale so any score from any scale can be converted into a z scale with z scores and this allows for really efficient communication across statisticians scientists can share data different researchers can share data if everything's converted to a z scale it's very easy to interpret and it's very simple calculation how do we convert or what i'll call a raw score to a z score it's just this formula right here we just take the raw score that's x subtract the mean m and divide by the standard deviation so we'll just take the raw score subtract the mean divide by the standard deviation that gives us a z score and if we do that for every score in a distribution then we can put our distribution on this common metric the z scale so what's also nice about a z scale or z scores is the mean z-score in any one sample is always going to be zero right so i take the raw score subtract the mean if my raw score is the mean so say body temperature the average body temperature in fahrenheit is 98.6 so 98.6 minus the mean assuming a normal distribution 98.6 minus 98.6 would be zero divided by whatever the standard deviation is would give me zero so the mean in any z-score distribution is going to be zero what's nice about that is if you have a negative z-score then i know you have your score is below average and if you have a positive z-score then i know you're above average so again let's look at this body temperature distribution here it is again in in degrees fahrenheit it's this nice normal distribution here it is again in celsius again nice normal distribution here it is in terms of z-scores again we did this in r and r sort of did the breaks a little bit uh differently for this graph but it's still a nice normal distribution and most importantly the average is zero so the average body temperature in terms of z scores is zero these are the same exact numbers from histogram to histogram histogram it's just in degrees fahrenheit degrees celsius or in z scores it's the same exact numbers we just converted them direct conversion so just to be clear let's assume we have a normal distribution of healthy individuals where the mean body temperature is 98.6 degrees fahrenheit and the standard deviation is about a half a degree so 0.5 suppose i pick one individual from that distribution at random and that individual's body temperature is 99.6 degrees now i want to convert that raw score that x 99.6 to a z-score well all i have to do is apply this z formula i take the raw score 99.6 subtract out the mean 98.6 divide by the standard deviation which was 0.5 so that's just one over point five or two so this individual z score is positive twoi've sort of been emphasizing how important it is to know what type of variable you're dealing with and emphasizing how important it is to look at your distributions in your variables it's also really important to keep in mind what scale of measurement you're dealing with so for all those histograms i presented the histograms first in degrees fahrenheit then in degrees celsius those are just two different scales of measurement to measure temperature right so i think we're all used to dealing with scales of measurement but in statistics what's nice is there's a standard scale and it's called the z scale so any score from any scale can be converted into a z scale with z scores and this allows for really efficient communication across statisticians scientists can share data different researchers can share data if everything's converted to a z scale it's very easy to interpret and it's very simple calculation how do we convert or what i'll call a raw score to a z score it's just this formula right here we just take the raw score that's x subtract the mean m and divide by the standard deviation so we'll just take the raw score subtract the mean divide by the standard deviation that gives us a z score and if we do that for every score in a distribution then we can put our distribution on this common metric the z scale so what's also nice about a z scale or z scores is the mean z-score in any one sample is always going to be zero right so i take the raw score subtract the mean if my raw score is the mean so say body temperature the average body temperature in fahrenheit is 98.6 so 98.6 minus the mean assuming a normal distribution 98.6 minus 98.6 would be zero divided by whatever the standard deviation is would give me zero so the mean in any z-score distribution is going to be zero what's nice about that is if you have a negative z-score then i know you have your score is below average and if you have a positive z-score then i know you're above average so again let's look at this body temperature distribution here it is again in in degrees fahrenheit it's this nice normal distribution here it is again in celsius again nice normal distribution here it is in terms of z-scores again we did this in r and r sort of did the breaks a little bit uh differently for this graph but it's still a nice normal distribution and most importantly the average is zero so the average body temperature in terms of z scores is zero these are the same exact numbers from histogram to histogram histogram it's just in degrees fahrenheit degrees celsius or in z scores it's the same exact numbers we just converted them direct conversion so just to be clear let's assume we have a normal distribution of healthy individuals where the mean body temperature is 98.6 degrees fahrenheit and the standard deviation is about a half a degree so 0.5 suppose i pick one individual from that distribution at random and that individual's body temperature is 99.6 degrees now i want to convert that raw score that x 99.6 to a z-score well all i have to do is apply this z formula i take the raw score 99.6 subtract out the mean 98.6 divide by the standard deviation which was 0.5 so that's just one over point five or two so this individual z score is positive two\n"