The following frequency table shows the Favorite Color data from our class survey.
Color Frequency
Black 2
Red 3
Blue 4
Green 6
Pink 3
Orange 2
Gray 1
Purple 1
White 2
Describe the variability for this data
One measure of variability is the range, highest value minus lowest. Since this is only a nominal scale, this is the only one that can be used.
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To describe the variability for this data, we can look at the spread or dispersion of the frequencies across different colors. One measure commonly used to describe variability is the range, which is the difference between the highest and the lowest frequency.
In this case, the highest frequency is 6 (for the color Green), and the lowest frequency is 1 (for the colors Gray and Purple). Therefore, the range is 6 - 1 = 5.
Another measure of variability is the interquartile range (IQR), which is the range of the middle 50% of the frequencies. To find the IQR, we first need to find the median of the frequencies. The median is the value in the middle when the frequencies are arranged in ascending order.
Arranging the frequencies in ascending order, we get: 1, 1, 2, 2, 2, 3, 3, 4, 6
The median is the middle value, which in this case is 2 (the 5th value).
Next, we need to find the 25th percentile and the 75th percentile. The 25th percentile is the value below which 25% of the frequencies fall, and the 75th percentile is the value below which 75% of the frequencies fall.
Since we have 9 values, the 25th percentile falls at the (25/100) * 9 = 2.25th value. Since this is not a whole number, we need to interpolate between the 2nd and 3rd values. The 25th percentile is calculated as 2 + (0.25 * (3 - 2)) = 2.25.
Similarly, the 75th percentile falls at the (75/100) * 9 = 6.75th value. Interpolating between the 6th and 7th values, we get 6 + (0.75 * (7 - 6)) = 6.75.
Therefore, the 25th percentile is 2.25 and the 75th percentile is 6.75. The IQR is the difference between the 75th percentile and the 25th percentile, which is 6.75 - 2.25 = 4.5.
These measures of variability, the range and the interquartile range, give us an idea of how spread out the frequencies are across different colors in our class survey.