Understanding centre and spread (year 13) Centre and spread are the two most basic concepts used for descriptive and comparative statistics. If we are describing one sample or comparing two samples, we want to be able to identify the one best number that describes the whole group (the centre) and how variable the individuals in that group are about that centre. Other descriptors like skew and unusual features are also useful, but the most basic information about the distribution is captured by describing centre and spread. centre spread one best number to describe the group how different members of the group are from each other position variation central tendency dispersion signal noise Centre Median and mean are measures of central tendency or average. If you needed one number to describe the whole group, this would be it. Centre describes position, how far along a scale a group is. Describe what you observe and use the median or mean as confirmation of your observations. Demonstrate that you understand what the mean/median measures in terms of the context (not the formula). Suitable words for demonstrating that understanding include “on average” and “tend to”. Spread As well as describing the centre or position of our sample, we want to describe its variability, how different the values are from each other, or how different the values are compared to the centre. We need something to measure the variability of the whole sample. IQR and SD are measures of variation or spread for a sample (or population). They describe how different the values are from each other. The discussion of spread should be separate from the discussion of centre, and should not include any reference to position along the scale. Range is not useful as a measure of spread, since it is determined only by extreme values. Describe what you observe and use the IQR/SD as confirmation of your observations. Give the IQR/SD with units in context and demonstrate that you understand what the IQR/SD measures in terms of the context (not the formula, so not “width of middle 50%”). The concept being described is the variability of the whole sample or population. Large values of IQR/SD indicate a lot of variability in the sample or population. What “large” means depends on the context. In manufacturing, variability needs to be small. Some natural populations are very variable. Consider whether the variation described by the IQR is large for the context you are investigating. If the data is approximately normally distributed, you could relate the SD to that distribution (showing statistical insight). You could combine your description of spread with your description and interpretation of the shape of the distribution. Note that a measure of variability is a measure for the whole group, so we say that “there is more variation in the heights of males than there is for females in my sample”. We don’t say “tends to” or “on average” when we are talking about variation. Choosing your statistic The mean is a more efficient measure of centre than the median (a sample mean tends to be closer to the population mean, on average, than a sample median is to the population median), so confidence intervals for means tend to be narrower than confidence intervals for medians. However, means are more affected by extreme values than medians, so for any context in which there are extreme values or a very skewed distribution, the median is a better measure of centre than the mean. Your exploratory data analysis will help you decide which measure you believe is best to use for an investigation. If using mean, use standard deviation with it. Both are affected by extreme values, so do not use these if your sample has very extreme values or is very skewed. Median and IQR are less efficient measures of centre and spread than the mean and SD, but they are robust (unaffected by extreme values). If using median, use IQR with it. Shift and Overlap Shift and overlap are comparisons of the centre relative to spread for two samples, answering the questions “Which one is bigger?” and “How much bigger, relative to the variation in each sample?” Think, describe what you see and relate it to the real world. You will not get credit for general sentences which do not relate to the context and could be taken from the table of statistics without understanding eg It is not acceptable to say that “the mean for females is 162.6cm which is about 5cm less than the mean for males at 167.5cm”, unless it is followed by further interpretation. Show understanding of the context (eg “taller” or “shorter” showing an understanding that you are discussing height) and understanding of the concept of average (eg “tend to”). Example 1: In my sample I notice that the year 9 boys tend to be taller than the year 9 girls. The middle 50% of the boys heights is shifted further up the scale than the heights of the girls. There is quite a lot of overlap between the middle 50% of boy and girl height, indicating that there are a lot of boys and girls in my sample who are quite similar in height. The mean height for boys in my sample was 167.5 cm, while the mean height for girls was 162.6cm. This confirms that the boys in my sample tend to be about 5cm taller than the girls. This makes sense because lots of year 9 boys I know are taller than the year 9 girls I know. In my sample I notice that the heights of year 9 boys have a similar spread to the heights of year 9 girls. This is confirmed by the sample statistics. The SD of my sample of girl heights is 10.7cm showing a reasonable amount of variation. The SD of boy heights is 9.8cm, only slightly smaller. This means that there is less variation in the heights of boys than girls in my sample. This makes sense because heights tend to be normally distributed, and the distribution of heights of both sexes in my sample are consistent with samples from a normal distribution with most heights in the middle and fewer taller and shorter people. In a normally distributed population almost everyone would be within 3SD of the mean. For males this would mean almost everyone would be between 138cm and 197cm tall which is true in my experience. For the females almost everyone would be between 130cm and 195cm which would also be true. Example 2 In my sample I notice that the males have driven at a faster maximum speed than females, on average. The middle 50% of male speed is shifted toward higher speed than females, and only part of the middle 50% of males and females overlaps. The median maximum speed driven by males is 120km/hr while the median maximum speed driven by females is 15km less at 105km/hr, which confirms my observations. In my sample I notice that the males seem to be more variable in the maximum speed driven compared to the female with more males driving at higher speeds. The IQR of maximum speed driven for females in my sample is 77.5 km/hr, which is a huge amount of variation in speed. For males the IQR was 40 km/hr, which is quite a lot less. This means that the males in my sample are more similar to each other in the maximum speed driven than the females are. This doesn’t at first make sense when I compare it to my initial observations. This does make sense when I look closer. The IQR being higher for females is because the female distribution is more bimodal with lots of females never having driven (maximum speed of zero), but most others clustered between 90 km/hr and 120 km/hr, so the statistical spread is higher for females than males.