In statistics, how do you find the interquartile range?
The following data represent the number of grams of fat in breakfast meals offered at McDonald’s 12, 23, 28, 2, 28, 33, 31, 11, 23, 40, 35, 1, 23, 33, 23, 16, 11, 8, 8, 17, 16, 15 Honestly, I don't want the answer. I just need to know how to go about getting to the answer. Like the formula or equation. I have read this section in my book 4 times and I am really confused.
Uses of interquartile range?
The interquartile range is a measure of spread or variation. Other measures of spread are the standard deviation and range. Outliers or data points that are very different from most of the numbers in the data set will cause the range (max value - min value) to indicate that the data is much more spread out than it actually is; the interquartile range is much more resistant to this problem. The standard deviation is not always a good measure of spread if the data has a distribution that is not close to a bell curve; the interquartile range is going to represent the spread better in cases where the data may be skewed (a bell curve where one side has a longer tail than the other). The interquartile range is not perfect, depending on the distribution it can be misleading by itself. It also does not have some mathematical properties that make the standard deviation useful in other calculations and hypothesis testing.
What is the interquartile range?
Interquartile range (IQR) is a measure of how the middle [math]50\%[/math] of data are spread around the median. If IQR is large, data are more spread out from the median, otherwise they are closer. In other words, discard the lower and upper [math]25\%[/math] of the sorted data and take the difference between the maximum and minimum value of the remaining, you will get IQR.Let’s say, you have some data like [math]A = \{5, 4, 7, 8, 3, 9, 6, 2, 1, 10\}[/math]. We will measure its IQR. After sorting the data we get [math]A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}[/math]. As the length of [math]A[/math] is even, split it to two equal halves. [math]B = \{1, 2, 3, 4, 5\}[/math] and [math]C = \{6, 7, 8, 9, 10\}[/math]. Now the median of [math]B[/math] (median of lower quartile of [math]A[/math], [math]Q_{1}[/math]) and [math]C[/math] (median of upper quartile of [math]A[/math], [math]Q_{3}[/math]) is [math]3[/math] and [math]8[/math] respectively. So, [math]IQR = Q_{3} - Q_{1} = 8 - 3 = 5[/math]For odd length data like [math]A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}[/math], split it to two equal halves discarding the middle element. For this case, [math]B = \{1, 2, 3, 4, 5\}, C = \{7, 8, 9, 10, 11\}[/math] and [math]Q_{1} = 3[/math] and [math]Q_{3} = 9[/math]. So, [math]IQR = Q_{3} - Q_{1} = 9 - 3 = 6[/math].IQR measures how well the mean can represent data. Besides, you can detect outlier using this. If any data is [math]1.5 \times IQR[/math] less than [math]Q_{1}[/math] or [math]1.5 \times IQR[/math] greater than [math]Q_{3}[/math], we call that as an outlier.
If interquartile range is defined as Q3-Q1, why do box and whisker plots show uneven whiskers above and below the median?
I know this is an old question, but I actually really like box and whisker plots, so I’m going for it.A box and whisker plot illustrates distribution, so the whiskers show the range of values for Q1 (negative whisker) and Q4 (positive whisker). These whiskers don’t originate from the median, but from the lowest point of Q2 and the highest point of Q3. They will only be equal in magnitude when the data is normally distributed (think a perfect Bell curve).Say we have the following values:1 1 1 1 1 3 3 3 5 6 7 8 9 10 11 11 11 11 12 20Dividing these values into quartiles gives usQ1: 1 1 1 1 1Q2: 3 3 3 5 6median: 6.5Q3: 7 8 9 10 11Q4: 11 11 11 12 20In this case, the negative whisker would stretch from 1 to 3, the lower box would stretch from 3 to 6.5 (the median), the upper box would stretch from 6.5 to 11, and the positive whisker would stretch from 11 to 20.However, the value 20 seems to be an outlier, so some people might exclude 20 from the analysis and represent it with an asterisk. In this case, the median would be 6 rather than 6.5, and so the negative whisker and boxes would not be much affected, but the upper whisker would stretch from 11 to 12, rather than from 11 to 20.Box plot review
What does a box and whisker plot represent?
A boxplot is a way of quickly visualizing a data set. It is based on the idea of putting all the data in order of size, then dividing up into chunks which are illustrated by the boxplot. The box itself stretches from the lower quartile to the upper quartile of the data. A quarter of the data has values that are less than the lower quartile. A quarter of the data has values that are greater than the upper quartile. The difference between the lower and upper quartiles is called the interquartile range or IQR. The median (the line across the box, not necessarily in the middle) is the middle value. Half the data is less than the median and half is greater than the median. The tips of the whiskers are the lower and upper adjacent values. The lower adjacent value is the smallest data value that is not more than 1.5 IQR below the median. The upper adjacent value is the largest data value that is not more than 1.5 IQR above the median. Any other data values that are too small or too large to be included in the whiskers are known as outliers and are indicated as stars or spots. Hope this helps.
In statistics, how do you compute the interquartile range?
The following data represent the number of grams of fat in breakfast meals offered at McDonald’s 12, 23, 28, 2, 28, 33, 31, 11, 23, 40, 35, 1, 23, 33, 23, 16, 11, 8, 8, 17, 16, 15 Honestly, I don't want the answer. I just need to know how to go about getting to the answer. Like the formula or equation. I have read this section in my book 4 times and I am really confused.
Why is the interquartile range halved in the calculation of quartile as a measure of variability/spread?
Why is the interquartile range halved in the calculation of quartile as a measure of variability/spread?It doesn’t have to be, it depends on what you want to represent. The semi-interquartile range is a sort of average distance from the centre of the distribution while the range is a sort of average total spread.You can take the words ‘average’ and ‘centre’ above with a pinch of salt, especially ‘centre’ unless the distribution is symmetrical.Perhaps the semi-interquartile range relates better to the standard deviation or the mean deviation (which really is the average deviation from a measure of the centre).
What is the difference between the lower quartile and the upper quartile?
Quartiles are basically location specifiers. The most common of these is the central or the second quartile commonly known as the Median. We define median as a point that divides the data set into two equal halves. Analogus to this definition the first or lower quartile divides the data set into two parts but in the ratio 1:3. Similarly the upper quartlie divides the data set in the ratio 3:1.For e.g10,14,17,19,21,25,27,29,35,38,42,45Lower quartile: 202nd quartile: 26Upper quartile: 32
What is quartile deviation?
Quartile deviation, also known as semi-interquartile range, is a measure of scatteredness (dispersion) of your data points - that is, whether the observations are densely clustered around a central value or are far apart from each other on an average. It is half the difference between the third and first quartiles.Quartile Deviation = [math]\frac{Q_3 - Q_1}{2}[/math]Quartile deviation prevents outliers from overly inflating the actual scatteredness of your data points.Say you ask someone to make a note of the daily average temperature (in °C) of a given city, over a certain week. Suppose they give you this:23.4 21.5 22.4 23.6 21.5 23.7 32.4All the readings are in the 21 - 23 range, except for the absurd 32.4. Something just doesn't look right in the data, does it? The temperature isn't suppose to fluctuate so widely in a span of a single week. Perhaps the person typed in the digits of those observations in a wrong order (that is, maybe he typed 32.4 instead of 23.4). Or maybe he didn't make any copying errors on his part, perhaps his source of information has misquoted the temperature. Or maybe the temperature actually was like that?You have no idea what happened, but you suspect that 32.4 is a suspicious value. This is where the quartiles come in. Instead of considering the highest and lowest values in your dataset to arrive at a measure of dispersion, you consider the bulk of the observations sandwiched somewhere in between them (because the highest value can be abnormally high if wrongly reported; similarly, the lowest value can be abnormally low if it were wrongly reported in the first place). You arrange your observations in increasing (or decreasing) order:21.5 21.5 22.4 23.4 23.6 23.7 32.4You identify the quartiles.21.5 21.5 22.4 23.4 23.6 23.7 32.4Here [math]Q_1[/math] = 21.5, [math]Q_2[/math]=23.4, [math]Q_3[/math]=23.7You work with only the data within [math]Q_1[/math] and [math]Q_3[/math] (both inclusive).21.5 22.4 23.4 23.6 23.7Now the data taken from the middle doesn't contain any blatantly absurd observations! This the essence behind using such a measure for dispersion based on the difference of the third and first quantiles of the original data.The Quartile Deviation this case is [math]\frac {23.7 - 21.5}{2} = 1.1[/math]
What is the best graph to illustrate ranges in a data series?
The minimum and maximum values are already visible in your graph, and for the median you could just add a small vertical line in each series. But there are some alternatives you might want to consider:A box-and-whisker plot (example from wikipedia)The thick lines in the box represent the median. The boxes visualize the interquartile range, and the hinges show the minimum and maximum value, but exclude outliers.This still doesn't show the density distribution, other graphical representations do, such as:The violin plot:Or the beanplot: