Descriptive Statistics, Part 2

welcome I'm Amanda raccoons and zap cue and in this tutorial we are going to talk about descriptive statistics in part 2 of this two-part tutorial we are going to look at measures of position including percentiles quartiles and standard scores and we're specifically going to focus on the standard score the z-score well then also going to briefly discuss how to calculate descriptive statistics in spss let's begin talking about specifically identifying defining and looking at examples of measures of position let's start with percentiles first now percentiles can be defined as values that divide a set of observations into 100 equal parts or points in a distribution below which a given percentile or P of cases lie let's look at an example of this let's say we have a distribution of scores and one of our scores is 33 and let's say this may be number of points that an individual scores on a test so we know that we have one individual in our distribution and they scored 33 points and we know that they their number of 33 is equal to the 50th percentile now if we look back at the definition of points in the distribution below which a given P of the cases lie we know that if a score is it within the 50th percentile then 50 percent of the scores are below the 50th percentile so we know this what then can we say about the score of 33 well we could say that 50% of individuals in our distribution scored below 33 let's take a look at another example and what percentiles are and how they're interpreted let's say that another person in our distribution scored a 73 and we know that the score of 73 is equal to the 75th percentile so again looking back at our definition what we know is that 75% of scores are below the 70th 75th percentile we could also look at it as 25% or above the 75th percentile so knowing this what then can we say about the score of 73 well what we could say is is that 75% of individuals scored below 73 and 25% of individuals scored above 73 okay now that we have an understanding of percentiles let's move on to quartiles quartiles are a rank rank order or they rank order the data into four equal parts the values that divide each part are called the first second and third quartile so they're denoted by q1 q2 and q3 respectively so in terms of percentiles what we can say is the quartet that quartiles can be defined as the as following for example quartile 1 is equal to the 25th percentile quartile 2 is equal to the 50th percentile or the median and quartile 3 is equal to the 70th 5th percentile so if we look at our score of 33 and we know that 33 is with is is equal to the 50th percentile or than the median we can then say that the score of 33 is equal to the second quartile or quartile 2 to see how that works so since we know that percent the 50th percentile is equal to the median the median is equal to the second quartile and 33 is equal to the 50th percentile we can say that 33 is equal to quartile 2 or the second quartile then let's look at 73 we know the score of 73 is equal to the 70th 5th percentile then what can we say it's Cortot what quartile is it equal to well as the 70th pretty fifth percentile is equal to the third quartile we can say that seventy-three equals the third quartile now that we have a basic understanding of percentiles and quartiles and as you can see arrow as I hope you're seeing what these are helping us do is understand the specific position of a score so it's not necessarily helping us understand the distribution as we as we as we looked at is look at Oh as we looked at whenever we were looking at measures of central tendency in dispersion so measures of central tendency and dispersion help us understand the overall distribution whereas percentiles and quartiles help us understand a specific score and how where it lies within that distribution so I'm hoping you're understanding that difference now before we move on let's go ahead and talk briefly about the standard scores standard scores are how many standard deviations a score or an element is from the mean so standard scores are important for the reason of convenience and comparability of different data sets and makes it easier to interpret and probably the most common standard score used is the z-score and that's what we're going to focus on next so as I said a z-score is a standardized score here we see the formula for a z-score first we see the formula for a population and then we see one for a sample the formula for the sample is Z is equal to X minus M and X denotes they value or the number that you're looking at divided by the standard deviation narrative Li a z-score specifies the precise location of each value within a distribution in relationship to the mean it gives us the number of standard deviations that a score lies either above or below the mean the sign of the z-score whether it's positive or negative then signifies whether the score is above the mean or below the mean so if a z-score is equal to zero that means that it is equal to the mean if the z-score is less than zero that means then it's less than the mean if a z-score is greater than zero then that means it's greater than the mean so if it if we have a z-score of 1 that means that the number that we're looking at or the value that we're looking at is one standard deviation greater than the mean whereas if we had a z-score negative two minus one let's say that would mean that the value that we're looking at is one standard deviation less than the mean if we had a z-score then of two weak would say that the value that we're looking at is two standard deviations greater than the mean and so on and so forth let's take a look at z-scores a little bit more practically in light of an example scenario let's consider a student's sample population of normally distributed educational statistics final exam scores that have a mean of 100 and a standard deviation of 15 if we know this what can we say about an individual who scores a hundred and thirty well we can say something about that score if we know its z-score and remember to calculate the z-score what we will do is we will take X minus the mean so 130 minus the mean of a hundred divided by the standard deviation of 15 and what we find out is is that the score of 130 has a z-score of +2 so what does the z-score of +2 mean well based on what we just talked about the score is two standard deviations above the mean now here it's important to note by convention outcomes that have z-score values less than minus 1.96 or greater than 1.96 are usually viewed as unusual or extreme in a normal distribution what we know is is that about 2.5 percent of the area lies below the z-score of minus 1.96 and 2.5 percent of the area lies above the z-score of plus 1.96 so together these two tails make the most extreme 5 percent of the outcomes in the distribution of scores that's why we would consider scores above a 196 unusual as well as scores below up a 1.96 unusual so going back to our example what can we say about our score of 130 that has a z-score of 2 well based on convention can we not say that this the person that scored 130 on the educational statistics file or final was extremely high let's consider extreme or unusual z-scores in light of another example let's consider extreme z-scores in terms of height we know that woman's height in the u.s. is approximately normally distributed with a mean of sixty-four inches in a standard deviation of two point five six inches now I want you to consider that you walk into a room of women and what you see is a woman who's 67 inches tall or 66 inches tall and a woman who is 71 inches tall okay so you see these two women which woman are you more likely to take note of the one that 66 inches or the one that's 71 inches you probably said the one that's 71 inches because we know that the average height of a woman is sixty-four inches so most of the women in the room are going to be around sixty-four inches and sixty-six is close to 64 however somebody ones not so close and chances are the woman who's 71 inches is going to look a lot taller than everyone else in the room in fact her height may be considered unusual or extreme now let's look at these two women's z-scores um and see if the if the conclusion we just made holds true here we'll see that the woman that was 66 inches if we calculate her z-score by taking her her inches - her or minus the mean divided by the standard deviation that her z-score is one point one seven and what we note by convention is this is an unusual or extreme we can then calculate the z-score for the woman that's 71 inches and what we note here is that her z-score is two point seven three and by convention that's above a 1.96 and therefore her height could be considered extreme now understanding z-scores can be helpful in many ways because z-scores really help us understand a person's position in a distribution we just looked at the example of height but let's look at how a z-score may be useful if we go back to our educational example an educator may want to know again may want to know how better understands let's say a specific students achievement score on multiple assignments in a course or on a specific assignment and of course examining z-scores for a student on each assignment or multiple assignments can inform the educator if a specific student is performing average or extremely higher extremely low and this can then inform their instruction so again a z-score helps inform the position of one score which is a little different than standard deviation mean which tells us about the entire distribution now that we understand z-scores let's move on and talk a little bit about how to calculate both them in other descriptive statistics and SPSS descriptive statistics can be calculated using three different functions or primarily three different functions in SPSS and that's frequencies descriptives and explore now there's some overlap in these functions in that they calculate mean and standard deviation however they do have different features for example frequency allows us to calculate specific percentiles whereas the explore function only Allah only calculates percentiles preset by the SPSS software and the frequency function also allows you to look at one variable at a time however the explore function enables you to examine a variable disaggregated by another variable so if I wanted to look at let's say course points disaggregated by gender or ethnicity I would use the explore function whereas if I was just interested in examining course points I might use the frequency function so I encourage you to explore the different functions in which you can calculate the descriptive statistics in SPSS and again those functions are frequency descriptives and explore if you're interested in calculating a z-score then you'll want to use the descriptive function so you'll go to analyze descriptive statistics and click descriptives once you've done that you'll choose what variable you want to analyze and below the variable list what you'll see is a little button that you can tick that says save standardized values as variables if you desire to calculate the Z scores for your entire sample population you tick this and then you tick okay so that's how you calculate z-scores and SPSS this no concludes part two of our tutorial on descriptive statistics you should now be able to identify and provide examples of different measures of physician including percentiles quartiles and z-scores and you should also understand how to calculate descriptive statistics and specifically z-scores in SPSS

Channel: The Doctoral Journey

Views: 48,873

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Keywords: statistics

Id: ZkEjYloGRIE

Length: 15min 20sec (920 seconds)

Published: Mon Aug 26 2013