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[h]SL 4.3 Measures of central tendency
[q] Averages
[a] Also known as central tendency like mean, median and mode.
[q] Mean
[a] The sum of the values, divided by the total frequency, i.e:
Mean = \(\frac{\sum_{i=1}^{n}x_{i}}{n}\)
[q] Median
[a] The middle of an ordered list. The \((\frac{n+1}{2})^{th }\) value.
[q] Mode
[a] The most frequent data value, or class (modal class).
[q] Mean and Frequency
[a] If you have 3’s in a lost, and were calculating the mean, you could do +3, +3, +3, +3, +3, ……., or add 3 times its frequecny (3×5). This tells us that, if we have a frequency table, we could get the overall sum by finding the sum of each valye multiplied by its frequecny, giving us the formula:
\(x̄=\frac{\sum_{i=1}^{n}f_{i}x_{i}}{n}\)
[q] Dispersion/Spread
[a] Instead of finding where the center of the data is, we are also interested in how spread out the dara is [ measures of dispersion]. We have seen a couple already, with range and IQR, but these are fairly basic measures.
[q] Standard Deviation
[a] This measures the square root of the avaergae difference from each point to the mean. i.e, \(S.D=σ = \sqrt{\frac{\sum(x_{i}-μ)^2}{n}}\), where \(μ\) = mean. But you will never have to do this manually in an exam.
[q] Variance
[a] Square of the standard deviation [\(σ^{2}\)].
[q] Constant Changes
[a] You may be asked what happens to the mean, s.d., variance if a basic operation (x,÷, +, -) was carried out on every data point. Note, any operation changes the mean in the same way, but a +/- just shifts the data, so doesn’t affect spread, and therefore, s.d.. A x/÷ also does the same operation to the differences to the mean, so therefore to the s.d. aswell.
[q] Quartiles w/GDC
[a] We saw how to manually find the quartiles of a list, and from a cumulative frequency table. Your GDC can also calculate these, and you must note that these results may differ from the manual figure.
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