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The normal distribution and standard deviations<br />Mean<br />-1<br />+1<br />In a normal distribution:<br />Approximately 68% of scores will fall within one standard deviation of the mean<br />
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The normal distribution and standard deviations<br />Mean<br />+1<br />-1<br />+2<br />-2<br />In a normal distribution:<br />Approximately 95% of scores will fall within two standard deviations of the mean<br />
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The normal distribution and standard deviations<br />Mean<br />+2<br />+1<br />-1<br />-2<br />+3<br />-3<br />In a normal distribution:<br />Approximately 99% of scores will fall within three standard deviations of the mean<br />
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Using standard deviation units to describe individual scores<br />Here is a distribution with a mean of 100 and and standard deviation of 10:<br />100<br />110<br />120<br />90<br />80<br />-1 sd<br />1 sd<br />2 sd<br />-2 sd<br />What score is one sd below the mean?<br />90<br />120<br />What score is two sd above the mean?<br />
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Using standard deviation units to describe individual scores<br />Here is a distribution with a mean of 100 and and standard deviation of 10:<br />100<br />110<br />120<br />90<br />80<br />-1 sd<br />1 sd<br />2 sd<br />-2 sd<br />1<br />How many standard deviations below the mean is a score of 90?<br />How many standard deviations above the mean is a score of 120?<br />2<br />
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Z scores<br />z scores are sometimes called standard scores<br />Here is the formula for a z score:<br />A z score is a raw score expressed in standard deviation units.<br />What is a z-score?<br />
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z-score describes the location of the raw score in terms of distance from the mean, measured in standard deviations<br />Gives us information about the location of that score relative to the “average” deviation of all scores<br />A z-score is the number of standard deviations a score is above or below the mean of the scores in a distribution.<br />A raw score is a regular score before it has been converted into a Z score.<br />Raw scores on very different variables can be converted into Z scores and directly compared.<br />What does a z-score tell us?<br />
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Mean of zero<br />Zero distance from the mean<br />Standard deviation of 1<br />The z-score has two parts:<br />The number<br />The sign<br />Negative z-scores aren’t bad<br />Z-score distribution always has same shape as raw score<br />Z-score Distribution<br />
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z = (X –M)/SD<br />Score minus the mean divided by the standard deviation<br />Computational Formula<br />
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Jacob spoke to other children 8 times in an hour, the mean number of times children speak is 12, and the standard deviation is 4, (example from text).<br />To change a raw score to a Z score:<br />Step One: Determine the deviation score.<br />Subtract the mean from the raw score.<br />8 – 12 = -4<br />Step Two: Determine the Z score. <br />Divide the deviation score by the standard deviation.<br />-4 / 4 = -1<br />Steps for Calculating a z-score<br />
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Using z scores to compare two raw scores from different distributions<br />You score 80/100 on a statistics test and your friend also scores 80/100 on their test in another section. Hey congratulations you friend says—we are both doing equally well in statistics. What do you need to know if the two scores are equivalent? <br />the mean?<br />What if the mean of both tests was 75?<br />You also need to know the standard deviation<br />What would you say about the two test scores if the SDin your class was 5 and the SDin your friend’s class is 10? <br />
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Calculating z scores<br />What is the z score for your test: raw score = 80; mean = 75, SD= 5?<br />What is the z score of your friend’s test: raw score = 80; mean = 75, S = 10?<br />Who do you think did better on their test? Why do you think this?<br />
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Transforming scores in order to make comparisons, especially when using different scales<br />Gives information about the relative standing of a score in relation to the characteristics of the sample or population<br />Location relative to mean<br />Relative frequency and percentile<br />Why z-scores?<br />
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Fun facts about z scores<br /><ul><li> Any distribution of raw scores can be converted to a distribution of z scores</li></ul>the mean of a distribution has a z score of ____?<br />zero<br />positive z scores represent raw scores that are __________ (above or below) the mean?<br />above<br />negative z scores represent raw scores that are __________ (above or below) the mean?<br />below<br />
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Figure the deviation score.<br />Multiply the Z score by the standard deviation.<br />Figure the raw score.<br />Add the mean to the deviation score.<br />Formula for changing a Z score to a raw score: <br /> X= (Z)(SD)+M<br />Computing Raw Score from a z-score<br />
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Standardizes different scores <br />Example in text:<br />Statistics versus English test performance<br />Can plot different distributions on same graph <br />increased height reflects larger N<br />Comparing Different Variables <br />
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How Are You Doing?<br />How would you change a raw score to a Z score?<br />If you had a group of scores where M = 15 and SD = 3, what would the raw score be if you had a Z score of 5?<br />
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Normal Distribution<br />histogram or frequency distribution that is a unimodal, symmetrical, and bell-shaped<br />Researchers compare the distributions of their variables to see if they approximately follow the normal curve.<br />
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Use to determine the relative frequency of z-scores and raw scores<br />Proportion of the area under the curve is the relative frequency of the z-score<br />Rarely have z-scores greater than 3 (.26% of scores above 3, 99.74% between +/- 3)<br />The Standard Normal Curve<br />
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Why the Normal Curve Is Commonly Found in Nature<br />A person’s ratings on a variable or performance on a task is influenced by a number of random factors at each point in time.<br />These factors can make a person rate things like stress levels or mood as higher or lower than they actually are, or can make a person perform better or worse than they usually would.<br />Most of these positive and negative influences on performance or ratings cancel each other out.<br />Most scores will fall toward the middle, with few very low scores and few very high scores.<br />This results in an approximately normal distribution (unimodal, symmetrical, and bell-shaped).<br />
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The Normal Curve Table and Z Scores<br />A normal curve table shows the percentages of scores associated with the normal curve.<br />The first column of this table lists the Z score<br />The second column is labeled “% Mean to Z” and gives the percentage of scores between the mean and that Z score.<br />The third column is labeled “% in Tail.”<br />.<br />
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Using the Normal Curve Table to Figure a Percentage of Scores Above or Below a Raw Score<br />If you are beginning with a raw score, first change it to a Z Score.<br />Z = (X – M) / SD<br />Draw a picture of the normal curve, decide where the Z score falls on it, and shade in the area for which you are finding the percentage.<br />Make a rough estimate of the shaded area’s percentage based on the 50%–34%–14% percentages.<br />Find the exact percentages using the normal curve table.<br />Look up the Z score in the “Z” column of the table.<br />Find the percentage in the “% Mean to Z” column or the “% in Tail” column.<br />If the Z score is negative and you need to find the percentage of scores above this score, or if the Z score is positive and you need to find the percentage of scores below this score, you will need to add 50% to the percentage from the table.<br />Check that your exact percentage is within the range of your rough estimate.<br />
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