Unit 5 1010
The mean of a distribution of scores is 57, with SD of 11. Calculate the standard error for a distribution of means based on samples of 35 people
OM = o/square root of N = 11/square root of 35 = 1.86
Why is range smaller for the means of samples of 10 scores than for the individual scores themselves?
Range is smaller for the means of samples of 10 scores than for the individual scores because the more extreme scores are balanced by lower scores when samples of 10 are taken. Individual scores are not attenuated in that way
Population mean is 5.382, with SD of 1.138, France has z score of 0.963. What is its actual happiness score?
- 0.963 = (X-5.382)/1.138 o X = 6.478
Characteristics of the Distribution of Means
- Distribution of means needs its own standard deviation (smaller) because it is less variable than the distribution of scores - U indicated that it is the mean of a population, M subscript indicates that the population is composed of sample means - the means of all possible samples of a given size from a particular population of individual scores - Symbol OM is for the SD of the distribution of means - the typical amount that a sample mean varies from the population mean o Subscript M stands for mean; this is the SD of the population of means calculated for all possible samples of a given size
Using z Scores to Make Comparisons
- Say you and friend both took a quiz, you got 92/100 the distribution of your class had mean of 78.1 and SD of 12.2 (different teachers) - Friend earned 8.1/10; distribution had mean of 6.8 with SD of 0.74 - Standardize scores in terms of respective distributions - Both you and friend scores about mean and have positive z scores - Your friend did better with respect to their class than you did with respect to yours
Why does the spread decrease when we create a distribution of means rather than scores?
- When we plotted individual scores, each extreme score was plotted on distribution but when we plotted means, we averaged each extreme score with two other scores - If we created a distribution of 10 scores rather than 3, the distribution would be even narrower because there would be more scores to balance out the occasional extreme o Larger the sample size, smaller the spread of the distribution of means
what does the normal curve do?
- allows us to determine the probabilities about data and then draw conclusions that we can apply - describes the distributions of many variables - as sample size approaches population size, distribution resembles normal curve
distribution of means
- distribution composed of many means that are calculated from all possible samples of a given size, all taken from the same population o The numbers that make up the distribution of means are not individual scores; they are means of samples of individual scores
transforming z scores into raw scores
- if we already know a z-score, we can reverse the calculations to determine the raw score - formula is same; just plug in all the numbers instead of X and solve algebraically
what shape of normal curve tells us
- normal curve is symmetric; 50% of scores fall below the mean and 50% above - approx 34% of scores fall between mean and z score of 1.0 and -1.0 - Approx.. 14% of scores fall between z scores of 1.0 and 2.0, and -1.0, -2.0 - Approx.. 2% of scores fall between z score of 2.0 & 3.0, and -2.0 & -3.0 o With addition, determine that approx. 68% (34+34) of scores fall within 1 SD or 1 z score of the mean o Approx. 96% (14+34+34+14) of scores fall within 2 SDs of mean and all fall within 3 - If you know you are about 1 SD below mean, you know you are in lower 50% of scores and 34% of scores fall between your score and the mean o By subtracting, find that 50-34=16% of scores fall below yours (16th percentile)
transforming raw scores into z scores
- only info we need to do this is the mean and standard deviation of the population of interest - ex. score on midterm is 2 standard deviations above the mean; z-score is 2.0 - if you fell exactly at the mean, z-score would be 0
if the mean on exam is 70 and the standard deviation is 10, and your score is 80, what is your z score?
- you are 10 point/1SD above the mean: z-score is 1.0
how to calculate a particular z-score
1. determine distance of particular person's score (x) from the population mean (u): X-u 2. express this distance in terms of standard deviation by dividing the population standard deviation (o)
how to convert z score to raw score (steps)
1. multiply the z score by the population standard deviation (o) 2. add the population mean to this product (u)
3 important characteristics of the distribution of means:
1. As sample size increases, the mean of a distribution of means remains the same 2. The standard deviation of a distribution of means ( the standard error) is smaller than the standard deviation of a distribution of scores. As sample size increases, the standard error becomes even smaller 3. The shape of the distribution of means approximates the normal curve if the distribution of the population of individual scores has a normal shape or if the size of each sample that makes up the distribution is at least 30 (the central limit theorem)
Three ideas about the normal curve help us to understand inferential statistics
1. The normal curve describes variability of many physical and psychological characteristics 2. The normal curve may be translated into percentages, allowing us to standardize variables and make direct comparisons of scores on different measures 3. A distribution of means, rather than a distribution of scores, produces a more normal curve
z-scores are useful because: (3)
1. They give us a sense of where a score falls in relation to the mean of its population (in terms of the standard deviation of its population) 2. Z scores allow us to compare scores from different distributions 3. Z scores can be transformed into percentiles
central limit theorem demonstrates 2 important principles:
1. repeated sampling approximates a normal curve, even when the original population is not normally distributed 2. a distribution of means is less variable than a distribution of individual scores
standardized z distribution allows us to do the following: (4)
1. transform raw scores into standardized scores called z scores 2. transform z scores back into raw scores 3. compare z scores to each other - even when the underlying raw scores are measured on different scales 4. transform z scores into percentiles that are more easily understood
standard normal distribution
A normal distribution with a mean of 0 and a standard deviation of 1.
Describe the process of standardization
In standardization, we convert individual scores to standardized scores for which we know the percentiles
What are the main ideas behind the central limit theorem?
It asserts that a distribution of sample means approaches the shape of the normal curve as sample size increases. Also asserts that the spread of the distribution of sample means gets smaller as the sample size gets larger
Explain what a distribution of means is
It is composed of many means that are calculated from all possible samples of a particular size from the same population
The mean of these 30 scores is 3.32. The SD is 0.69. Using symbolic notation and formulas, determine the mean and standard error of the distribution of means computed from samples of 10
Mean of distribution of means will be the same as mean of the individual scores Standard error will be smaller than SD; we must divide by the square root of the sample size of 10: OM = O/square root of N = 0.69/ square root of 10 = 0.22
What do the numeric value and sign (negative or positive) of a z score indicate?
Numeric value tells us how many SDs a score is from the mean of the distribution, the sign tells us whether the score is above or below the mean (+ z-score indicates the raw score is higher than the mean average, - z-score reveals the raw score is below the mean average; ex, z-score is equal to -2, it is 2 standard deviations below the mean)
Central limit theorem
The theory that, as sample size increases, the distribution of sample means of size n, randomly selected, approaches a normal distribution
What does it mean to say that the normal curve is unimodal and symmetric?
Unimodal means there is one mode or high point in the curve. Symmetric means the left and right sides of curve have the same shape and are mirror I mages of each other
z distribution
a normal distribution of standardized scores
The mean of a population is 14 and the SD is 2.5. Using the formula, calculate z scores for the following raw scores: 11.5, 18
a. 11.5; (14-11.5)/2.5 - z score = 1.0 b. 18; (14-18)/2.5 - z score = -1.6
The mean of a population is 14 and the SD is 2.5, calculate raw scores for the following z scores: 2, -1.4
a. 2; = 2(2.5) + 14 - raw score = 19 b. -1.4 = -1.4(2.5) + 14 - raw score = 10.5
z statistic
an inferential statistic used to determine the number of standard deviations in a standard normal distribution that a sample mean deviates from the population mean stated in the null hypothesis
standard error
name for the standard deviation of a distribution of means
z score
number of standard deviations a particular score is from the mean
normal curve
specific bell-shaped curve that is unimodal, symmetric and defined mathematically
standardization
way to convert individual scores from different normal distributions to a shared normal distribution with a known mean, standard deviation, and percentiles
