Remember this? We know the percentages that fall within the various portions of the normal distribution of z scores


 Roderick Moore
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1 More on z scores, percentiles, and the central limit theorem z scores and percentiles For every raw score there is a corresponding z score As long as you know the mean and SD of your population/sample to convert it! For every z score, there is a corresponding percentile So, if you want to find the percentile for a score, you would convert it to a zscore first, then find the particular percentile What is a percentile? Percentile rank The percentage of individuals with scores below the value IQ of 100 = what percentile rank? percentile rank of 50% Percentile Not a percentage IQ of 100 = what percentile? 50 th percentile Remember this? We know the percentages that fall within the various portions of the normal distribution of z scores 2.5% 34% 34% 13.5% 13.5% 2.5% How can we get more specific percentages? A z table! What does the z table tell you? 45.05% from between the mean and my score 4.95% above my score
2 So, what is the percentile for my z score of 1.65? 45.05% above the mean; 50% below the mean = 95.05% of the scores fall below a z score of 1.65 What about a z score of 1.65? z tables only include the positive z scores Because z distributions are symmetrical they only need to include one half of the distribution Here 45.05% of the scores are between my z score (1.65) and the mean Let s graph it Visualizing a z table z tables give you this percentage Working with a z table 45.05% When we ask for percentile, we are asking for what percentage of scores are under our score 50% % = 4.95% When given a z score Calculate the percentile rank for each of these Calculate the percentage of scores above it (for each) What percent of scores are more extreme than this score? z =.74 z = .85 z = Try this one SAT math scores are known to have means of 500 and SD of 100 Annie scored a 620 on the math portion of the SAT Calculate the percentile rank for Annie s score Calculate the percentage of scores above it What percent of scores are more extreme than this score?
3 Central limit theorem So far, we are talking about distributions of scores within populations (or samples) In the behavioral sciences, we rarely ask questions about individual subjects or scores Instead we ask questions about a sample of scores we have obtained in comparison to a population The goal of inferential statistics To infer information about a population using a sample Why to do this? We can describe a sample this way Is it an extreme sample? Is it an average sample? Remember sampling error? The amount of error between a sample statistic and a population parameter Statistic: describes a sample Parameter: describes a population Take some samples of the heights for students in this class 3 scores in each sample Give me the means Let s graph the means Compare the distribution of scores to the distribution of the sample means Notice there are some differences here How would you describe the distribution? The distribution of means looks normal (so long as you have enough means sampled) What about the variability? The distribution of means has less variability than the distribution of scores Central Limit Theorem A distribution of sample means approaches a normal curve as sample size increases Has the same mean as a distribution of scores from the population Has a smaller standard deviation We call the standard deviation for a sample of means standard error Does it make sense to you why the standard deviation for means (standard error) is smaller than the standard deviation for scores?
4 Formula for standard error N = the size of the sample drawn Notice that when N = 1, we are describing a score (a sample of 1) and the SD of the means is just the SD of the population The central limit theorem, in pictures Notice that the larger our sample, the more our distribution of means becomes a normal distribution If we got to 30 and above, that is typically our cutoff for saying it will be normal How to use this information? First let s start by looking back at an individual score and describing how extreme it is SAT scores are known to have a mean of 500 and a SD of 100 A Student from Whittier High School scored 520 on the 2010 SAT How much better did this student do than most other students? Notice that we are asking about a single score here We did this last lecture Calculations: for the individual z= ( ) / 100 =.2 Go to the z table and find.2 It is higher than 57.93% of scores It is NOT a very extreme score What would happen if 100 students took the SAT at Whittier High School and averaged 520? Is this an extreme sample? Here we are given: Population mean = 500 Population SD = 100 For a distribution of means The mean is the same µ M = µ = 500 We use standard error instead of standard deviation (because it is a distribution of means) σ M = σ / sqrt(n) = 100 / sqrt (100) = 10 Calculate the z statistic for the mean = 20 / 10 = 2.00 How good is that z score? Holy crap! These students scored better than 97.72% of other samples (of 100 students) taken from the population Does this make sense? Finding one student that scores higher than the mean is not that impressive Especially since 520 is not even a full standard deviation above the mean BUT finding 100 students that in total average that much higher than the mean is rare
5 IQ example The mean and SD for IQ in the general public are 100 and 15 respectively Two groups of 10 students are sampled from Rio Hondo College The average IQ score of group 1 was 102 The average IQ score of group 2 was 105 Based on the data presented here, what percentage of population samples would have higher IQ s than each of our samples What percentage of population samples would have lower IQ s than each of our samples?
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