Chapter 1: Statistical Analysis
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Define error bars
Error bars are a graphical representation of the variability of data to show the range of data or the standard deviation
If time is measured using an analog stopwatch, what is the error or uncertainty for recorded measurements?
Analog stopwatches have an error or uncertainty of ±1 standard deviation
Identify areas that are not included in determining experimental uncertainty
Uncertainty does not account for faulty equipment, human error or invalid assumptions
Describe how random error can be reduced when conducting an experiment.
Random errors occur in a non-reproducible way. In an experiment, random error can be reduced by repeating the measurement, taking the average and using the variation from the mean to show the reproducibility of the measurements.
Distinguish between accuracy and precision when taking measurements.
The accuracy of a measurement refers to its 'closeness' to the 'true' value. In an experiment, accuracy is taken as the difference between the average of the measured results and the 'true' value of the quantity. Precision in an experiment refers to the range of values in the results and the reproducibility of measurement.
Discuss how systematic error can affect accuracy and precision.
Systematic error can occur in several ways, e.g. in the equipment, the way the equipment was used or by the observer making the measurement. Systematic errors can be caused when instruments are not used under normal conditions, e.g. temperature and humidity can affect measuring devices. If the instrument is used correctly in adverse conditions, measurements can have high precision but reduced accuracy. Systematic errors occur if the instrument is not correctly zeroed, or the observer consistently misreads the instrument e.g. reads the wrong part of the meniscus. Accuracy is low, but results can have high precision.
what is meant by the mean?
The mean is the sum of all the values divided by the number of values (often called the average).
What is meant by the standard deviation?
The standard deviation summarises the spread of data around the mean.
In statistics, what is meant by frequency and frequency distribution?
In statistics, frequency refers to the number of times a value occurs and is called the frequency of that value. Frequency distribution can be shown with a table or a graph to show how often that value occurs compared to other values.
Distinguish between a discrete variable and a continuous variable
A discrete variable has exact values, e.g. number of test tubes set up, while a continuous variable can have any value in a given range, e.g. temperature.
What is the Gaussian curve?
The Gaussian curve is a bell-shaped curve which has a centrally located mean. IT is the normal distribution curve.
The diagram shows a normal distribution curve. In many biological situations, where there are large numbers of values, the frequency graph shows this shape. Identify useful features of the normal curve.
The normal distribution curve is symmetric about the mean and the range of values is unlimited. In the normal distribution curve, 68% of all values are withing 1 standard deviation of the mean. If using the z-score, this means that 68% of values have a Z-score between 1 and -1. The normal distribution curve also shows that about 95% of all values are within 2 standard deviations of the mean. This means that it is highly likely that a value is within 2 standard deviations of the mean. The normal curve also shows that 99.7% of values lie within 3 standard deviations of the mean. This infers there is a very high probability that any value is within 3 standard deviations of the mean.
What is the Z-score?
The Z-score is a standard score that states how many standard deviations a raw score is from the mean.
What is meant by variance and state the symbols used for sample variance and population variance.
The variance of a set of data is the square of the standard deviation. To distinguish between the standard variation of a population and the standard deviation of a sample drawn from this population, sample variance is given the symbol s2 and population variance is given the symbol σ2.
Explain the use of standard deviation when using several samples of data.
The standard deviation is useful to show how far a set of values is spread away from the centre. This is important when comparing different sample of data. A small standard deviation can infer high precision in measuring techniques and other aspects of experimental procedure and results can also be inferred by comparing the standard deviations of different sets of data.
Outline why the Z-score is often called a test statistic?
The Z-score is often used in tests of hypotheses and because of this important role it is often called a test statistic.
Outline the usefulness of plotting collected data.
Plotting data on a graph provides a visual means of showing the relationship between the dependent variable and the independent variable.
Sometimes you need to compare two sets of data to see if there is a correlation. Outline how you would find the relationship between two sets of variables.
To show the relationship between two sets of variables, you can draw a scattergram and plot each point.
When designing an experiment, an initial step involves stating the experimental hypothesis. What is meant by a two-tailed experimental hypothesis?
An experimental hypothesis is an educated guess about what will happen if a particular condition changes. A two-tailed experimental hypothesis will look at both sides of the mean (i.e. both tails of the distribution).
When looking at results, the level of significance needs to be determined, that is, the results are significant and probably not due to chance. When are results considered to be due to chance?
If there is a 5% of less probability, then the results are considered to have occurred due to chance rather than changes in the independent variable.
Outline why correlation does not imply causation. Use an example in your response.
Correlation between two sets of variables may mean one may cause the other, however the correlation method does not confirm causality. Experimental investigations are needed to study cause and effect. In an experiment, the independent variable can be manipulated and extraneous variable controlled to discover how the dependent variable changes in different conditions. Tests of correlation, e.g. Spearman's rho determine linear relationships. For example, there could be a high positive correlation between teenagers that watch the news each night on TV and those who own a pet dog. This does not mean that watching this nightly news causes teenagers to own a dog, or owning a dog causes teenagers to watch the nightly news.
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