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Analysis of Quantitative Data CENTRAL TENDENCY DISPERSION TABLES GRAPHS Analysis of quantitative data Specification: To analyse quantitative data including measures of central tendency (mean, median and mode), measures of dispersion (range and standard deviation) and graphs (bar charts and frequency tables) Central tendency: the tendency for the values of a variable to cluster round its mean, mode, or median. Measures of central tendency Descriptive statistics include measures of central tendency which are mode, median and mean average Data analysed in such a way that it is clearly displayed and understood (tables and graphs) Examples: Mode: The most common score in a set of scores 1 5 7 8 8 12 12 12 15 (mode is 12) Median: The middle score in a set of scores (in order) 1 5 7 8 8 12 12 12 15 – median is 8 1 5 7 8 8 11 13 14 15 20 – median is 9.5 (8+11)/ 2 =9.5 Mean : arithmetical average (adding all the scores in the set and dividing by the numbers of scores in a set) 1 5 7 8 8 11 13 14 15 20 – mean is 10.2 Mean Strengths Mean Limitations Median Strength Median Limitation Mode Strengths Mode Limitations Practice! For each of the following sets of data (a) calculate the mean, (b) calculate the median, (c) calculate the mode 2, 3, 5, 6, 6, 8, 9, 12 ,15, 21, 22, 80 2, 2, 4, 5, 5, 5, 7, 7, 8, 8, 8, 10 2, 3, 8, 10, 11, 13, 14, 14, 29 Extension: Which measure of central tendency is more appropriate for each set of data? And why? Measures of dispersion (Range and SD) Calculates the spread of score in a data set Range – The range is a measure of dispersion, found by finding the highest score/number and taking away the lowest score giving the difference between the two. 5 7 8 8 12 12 12 15 – range is 10 (15-5) Influenced by extreme scores so it may not always be useful Doesn’t tell us if the scores are bunched around the mean score or more equally distributed 1 7 8 8 12 12 15 16 55 (55-1 = 49) Range Strength Range Limitations Standard Deviation A normal distribution of data means that most of the examples in a set of data are close to the average, while relatively few examples tend to one extreme or the other. SD calculates the distance of a score from its group mean. The example here shows the SD of IQ. Each shaded block is 1 SD away from the mean. Measures of dispersion Standard deviation Measure of how far scores vary (deviate) from the mean average. The higher the standard deviation (SD), the greater the spread of scores around the mean value (the data is more varied) A low SD indicates that the scores tend to be closer to the mean of the set (data is less varied) The lower the SD – the more accurate and representative the mean of the data set is. Add up the differences squared for all the scores and then divide that number by the number of scores minus 1. Then finally, find the square root and you have the standard deviation Task Using the example on page 50 (on the next slide) and steps 1-5 Calculate the standard deviation for these scores: Score (x) Mean(x̄ ) Deviation (x-x̄ ) Squared deviation (x-x̄ )2 6 9 4 8 3 Step 4: n-1= ? (n: number of scores) Extension: Help others if you finish you calculation Sum of deviations squared: Standard deviation: ______ Standard deviation The SD does not necessarily indicate a right or wrong value. It is a descriptive statistic – it describes the distribution in relation to the mean. Another way of looking at Standard Deviation is by plotting the distribution as a histogram of responses. A distribution with a low SD would display as a tall narrow shape, while a large SD would be indicated by a wider shape Explain why a researcher may choose to use SD over the range to measure dispersion? Standard deviation Strengths SD is the most sensitive measure of dispersion as it is derived by using every score in the data set It is a more precise measure of the spread of data from the mean as compared to the range Limitation SD can be affected by extreme values (outliers) Frequency tables A frequency table records the number of times a score is found, rather than the score itself being displayed against each participant Useful as the distribution can be seen in table form. A histogram or frequency graph can be used to display the data Table to show frequency of self rated obedience scores Self reported obedience scores Frequency 1 2 2 5 3 4 4 2 5 3 Histogram/frequency graph This type of graph is used to present the distribution of the scores. Unlike a bar chart, where the bars a separated by a space, the bars on a histogram are joined to represent continuous data rather than categorical data. The values are presented on the x-axis and the height of each bar represents the frequency of the variable TASK Sketch a histogram based on the data in the previous slide Summary tables Summary tables represent measures of central tendency and dispersion clearly Copy out the table Describe the features/trends of the data: Table showing the self-rated obedience scores of males and females (How obedient the participant thinks they are) Males Females Mean obedience rating 4.1 7.4 Median obedience rating 4 7 Mode obedience rating 4 6,7,9 Range of obedience ratings 3 3 Standard deviation 1 1.2 Bar graph Useful to illustrate summary data Bar chars are used to present data from a categorical variable such as the mean, median or mode. Categorical value is placed on the x axis and the height of the bars represent the value of the variable. TASK Sketch a bar graph A bar graph showing the mean self rated obedience scores (from the table on the previous slide) All graphs need a main title and axis also need titles Include a space between each bar (for bar graphs only)
