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topic 8
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- 12 main ways to estimate the centre of the distribution of data1. sample median
2. sample mean - 2mean- the average
- most frequently used measure of central tendency
- sum of all scores divided by total number of scores in the dataset
- if the distribution is skewed, the mean will be pulled in the direction of the tail - 3robustness- the mean is very susceptible to the presence of outliers so is not robust
- the median is very robust, as an outlier is unlikely to effect as much - 4advantages of using the mean- takes into account information from outliers, where the median wastes it - therefore more powerful than the median
- has a rich theory behind it, the central limit theorem which makes it useful - 5disadvantages of using the mean- does not carry meaningful quantitative information from nominal or ordinal scales
- sensitive to extreme values, this is why it is important to visualise your data first - 6variance- measures how much each observation deviates from the mean
- larger deviation = greater variability
- deviations are +ve if they are above the mean, and -ve if below
- the sum of squared deviations from the mean is averaged by n-1 where n is the number of samples to give a better measure of spread
- the sample mean can be used to estimate the population mean, and it is unbiased as it is equally likely for the sample mean to be larger or smaller than the actual population mean. Dividing it by n-1 ensures the sample variance is also an unbiased estimate of the population variance, making it a useful measure of spread
- the number of independent information is called the degrees of freedom
- the variance is important, but is not the best measure for summarising data, as the units have been squared so will be different from the original data. To overcome this, we take the squareroot of the variance, which is the sample SD - 7normal distribution- sample mean and sample SD can give a measure of the centre and spread of a distribution
- the normal distribution is bell shaped and described by the mean and SD
- benefits of using the normal distribution are that we can use it to:
- describe distributions of observations
- describe the distribution of statistics, such as the sample mean (this is what makes it so valuable) - 8describe the 68-95-99.7 ruleAreas under the curve are used to calculate probabilities from the distribution
- in a normal distribution, 68% of sccores will be 1 SD +/- the mean
- 95% will fall within +/- 2 SDs
- 99.7% will fall between +/- 3 SDs - 9mean of means- can summarise variability by calculating the mean of however many sample mens you have taken
- calculating the sample mean does not give the true population mena
- finding the mean of more samples will be more reliable for describing the population than a single observation
- the sample mean has its own distribution, which is different to the population distribution
increasing sample size narrows the distribution and lowers the SD
- the distribution of sample means gets more symmetrical the larger the sample size taken, even though the population we are describing may be skewed - this is the central limit theorem - 10confidence intervals- the sample mean has a roughly normal distribution
- in any normal distribution about 95% of observations occur within 2 SDs of the mean
- therefore in 95% of samples the sample mean will be within 2 SD of the population mean
- this is called a 95% confidence interval for the population mean. The value of 2 x the SD = the margin of error of the interval
- we use confidence intervals to make conclusions based on a random sample - 11the null hypothesisis the hypothesis to be tested - usually the opposite of our research question since it is a hypothesis of no difference and is denoted by H0
- a hypothesis test aims to decide if we can reject H0, or if it should be retained
- if it is not rejected, the interpretation is that the data are consistent with a null hypothesis of no difference - 12alternative hypothesisif the null hypothesis is rejected, infers that the data is available to provide evidence for the alternative hypothesis (H1) which is often the research hypothesis
- 13P-valuesP-values help you decide if the H0 or H1 is correct
- the P-value is the probability of the outcome you observe if it is a fair test
- a common threshold for deciding between the null hypothesis and alternative hypothesis is 0.05
- if the p-value is less than 0.05, you can say there is evidence of an effect at the 5% level - 14one sample t-test- estimates the SD of the distribution of the sample mean for the given sample size and variability in our data
- this estimated SD of the sample mean is called the standard error of the mean and tells you how accurate your sample mean is going to be in practice
- after calculating the standard error, the t statistic is how many standard errors the observed sample mean is from the hypothesised population mean - anything more than 3 standard errors is going to be pretty unlikely
- software takes the t statistic and calculates a p-value for our test - 152 assumptions of one-sample t tests1. normality: the distribution should be reasonably symmetric, or the sample size should be large enough that the central limit theorem says the sample mean will have a normal distribution anyway
2. independence: the method by which the data was collected or measured must be independent from eachother. This assumption is established by appropriate methodology and design, along with the inherent nature of the variables chosen - 16type 1 errors- false positives
- rejecting the null hypothesis when it is actually true
- deciding there is an effect when there is really no association - 17type 2 errors- false negatives
- accepting the null hypothesis when the alternative hypothesis is true
- failing to observe an effect when there is one - 18power- it is easy to make fewer type 1 errors by lowering the significance threshold - however you may then make more type 2 errors by making it too hard to reject the null hypothesis
- usually fix a standard significance threshold such as a=0.05 or a=0.001 and look at what might affect the changeof a type 2 error
- the power is the probability of not making a type 2 error
- we can improve the power of a study by increasing sample size, decreasing variability in the response, increasing the effect size you are trying to detect
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- HRSS3101 Karteikarten am topic 8, erstellt von Soph B am 23/06/2020.
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