Representing Data - Statistics 1
Averages
The mean, mode and median are types of averages. They are different methods for working out he average for the data. If the data is skewed then it is best to use the median for the average as it will be the most accurate. The mean is affected by extreme values and the number that is the mode is just by chance and therefore not the best average to use.
Spread of Data
The range, interquartile range, variance and standard deviation all measure the spread of the data. The larger these values are the more spread out the data is, the smaller these values are the less spread out the data is. If the data is skewed then it is best to use the interquartile range as it will be the most accurate. The range, variance and standard deviation will be affected by extreme values.
Frequency Table
The height of 100 people, chosen at random, was measured to the nearest centimetre and the data is summarised below.
The mean, mode and median are types of averages. They are different methods for working out he average for the data. If the data is skewed then it is best to use the median for the average as it will be the most accurate. The mean is affected by extreme values and the number that is the mode is just by chance and therefore not the best average to use.
Spread of Data
The range, interquartile range, variance and standard deviation all measure the spread of the data. The larger these values are the more spread out the data is, the smaller these values are the less spread out the data is. If the data is skewed then it is best to use the interquartile range as it will be the most accurate. The range, variance and standard deviation will be affected by extreme values.
Frequency Table
The height of 100 people, chosen at random, was measured to the nearest centimetre and the data is summarised below.
Height (cm) |
Frequency |
160 – 169 |
40 |
170 – 179 |
55 |
180 – 190 |
5 |
If the data is measured to the nearest cm then you first need to work out the correct class widths:
Height (cm) |
Frequency |
159.5 – 169.5 |
40 |
169.5 – 179.5 |
55 |
179.5 – 190.5 |
5 |
Mean
Work out the sum of the midpoints of the correct class width multiplied by its corresponding frequency and then divide by the total frequency. This will be an estimate for the mean as the exact values of the data are unknown and you are assuming they are the midpoint.
Work out the sum of the midpoints of the correct class width multiplied by its corresponding frequency and then divide by the total frequency. This will be an estimate for the mean as the exact values of the data are unknown and you are assuming they are the midpoint.
Standard Deviation
You need to work out the variance first and then square root it to get the standard deviation.
You need to work out the variance first and then square root it to get the standard deviation.
Mode
It is not possible to work out the mode as the exact values of the data are unknown. However you can work out the modal class width. This will be the class width with the highest frequency.
Modal class width = 169.5 – 179.5
Median, Lower Quartile and Upper Quartile
To work out the lower quartile Q1, Median Q2, and upper quartile Q3 you need to use linear interpolation. You need to work out the quarter value, the middle value, and the three quarter value for the amount of numbers in the data set which is the total frequency. Depending on your exam board, you will either use n/4, n/2 and 3n/4 respectively OR (n+1)/4, (n+1)/2 and 3(n+1)/4. In this example, I have used the first equations.
Lower quartile value = 100/4 = 25
Middle value = 100/2 = 50
Upper quartile value = 3(100)/4 = 75
You then need to work out the cumulative frequency to work out the class width each value lies in.
It is not possible to work out the mode as the exact values of the data are unknown. However you can work out the modal class width. This will be the class width with the highest frequency.
Modal class width = 169.5 – 179.5
Median, Lower Quartile and Upper Quartile
To work out the lower quartile Q1, Median Q2, and upper quartile Q3 you need to use linear interpolation. You need to work out the quarter value, the middle value, and the three quarter value for the amount of numbers in the data set which is the total frequency. Depending on your exam board, you will either use n/4, n/2 and 3n/4 respectively OR (n+1)/4, (n+1)/2 and 3(n+1)/4. In this example, I have used the first equations.
Lower quartile value = 100/4 = 25
Middle value = 100/2 = 50
Upper quartile value = 3(100)/4 = 75
You then need to work out the cumulative frequency to work out the class width each value lies in.
Height (cm) |
Frequency |
Cumulative Frequency |
159.5 – 169.5 |
40 |
40 |
169.5 – 179.5 |
55 |
95 |
179.5 – 190.5 |
5 |
100 |
The 50th value lies in the second class width, the 25th value lies in the first class width and the 75th value lies in the second class width. Then to work out the values use the following formulae:
Interquartile Range
Interquartile Range = upper quartile – lower quartile
Interquartile Range = 176 – 166 = 10cm
List of numbers without a frequency table
Data: 3, 3, 4, 4, 5, 5, 5, 6, 6, 9
Mean
Add up the values and divide by the amount of numbers.
Interquartile Range = upper quartile – lower quartile
Interquartile Range = 176 – 166 = 10cm
List of numbers without a frequency table
Data: 3, 3, 4, 4, 5, 5, 5, 6, 6, 9
Mean
Add up the values and divide by the amount of numbers.
Standard Deviation
You need to work out the variance first and then square root it to get the standard deviation.
You need to work out the variance first and then square root it to get the standard deviation.
Mode
It is the most common number in the data set. If two numbers are equally common then both numbers will be the mode.
Data: 3, 3, 4, 4, 5, 5, 5, 6, 6, 9
Mode = 5
Median, Lower Quartile and Upper Quartile
First make sure the data is listed from smallest to largest. To work out the lower quartile Q1, Median Q2, and upper quartile Q3 you need to use linear interpolation. You need to work out the quarter value, the middle value, and the three quarter value for the amount of numbers in the data set which is the total frequency.
Data: 3, 3, 4, 4, 5, 5, 5, 6, 6, 9
It is the most common number in the data set. If two numbers are equally common then both numbers will be the mode.
Data: 3, 3, 4, 4, 5, 5, 5, 6, 6, 9
Mode = 5
Median, Lower Quartile and Upper Quartile
First make sure the data is listed from smallest to largest. To work out the lower quartile Q1, Median Q2, and upper quartile Q3 you need to use linear interpolation. You need to work out the quarter value, the middle value, and the three quarter value for the amount of numbers in the data set which is the total frequency.
Data: 3, 3, 4, 4, 5, 5, 5, 6, 6, 9
Interquartile Range
Interquartile Range = upper quartile – lower quartile
Interquartile Range = 6 – 4 = 2
Stem and Leaf Diagrams
To work out the Mean, Standard Deviation, Mode, Median, Q1, Q2, Q3 and the IQR, you will use the same methods as for a list of numbers without a frequency table with the numbers in the stem and leaf diagram.
Box Plots
The only values you can work out from a Box Plot are the Interquartile range, the range, and the Median. You can read what the Median is from the Box Plot. You can read the upper quartile and lower quartile values from the box plot and you can subtract them to work out the interquartile range. You can read the highest and lowest values from the box plot and you can subtract them to work out the range.
Histograms
Histograms are drawn from frequency tables therefore you can use the frequency table to work out the Mean, Standard Deviation, Mode, Median, Q1, Q2, Q3 and the IQR. The class widths will be the widths of the rectangles and height of the rectangles will be the Frequency Density. Fd = class width / frequency. The area of the rectangles will represent frequency since frequency = Fd x class width. The height and width of the rectangles can be in different ratios. The Fd values can be times by 10 and the class widths can be times by 2 before the Histogram is drawn.
Cumulative Frequency Graphs
You draw cumulative frequency graphs from frequency tables. You use the upper limit of the class widths as the x-coordinate, and you use the cumulative frequency as the y-coordinates for the points that you plot. You need connect the points with a smooth curve. The lower limit of the first class width will be where the graph starts from on the x-axis.
Frequency Polygons
You draw frequency polygons from frequency tables. You use the middle value of the class widths as the x-coordinate, and you use the frequency as the y-coordinates for the points that you plot. You need connect the points with straight lines. The frequency polygon will start from the first point and end on the last point plotted.
Interquartile Range = upper quartile – lower quartile
Interquartile Range = 6 – 4 = 2
Stem and Leaf Diagrams
To work out the Mean, Standard Deviation, Mode, Median, Q1, Q2, Q3 and the IQR, you will use the same methods as for a list of numbers without a frequency table with the numbers in the stem and leaf diagram.
Box Plots
The only values you can work out from a Box Plot are the Interquartile range, the range, and the Median. You can read what the Median is from the Box Plot. You can read the upper quartile and lower quartile values from the box plot and you can subtract them to work out the interquartile range. You can read the highest and lowest values from the box plot and you can subtract them to work out the range.
Histograms
Histograms are drawn from frequency tables therefore you can use the frequency table to work out the Mean, Standard Deviation, Mode, Median, Q1, Q2, Q3 and the IQR. The class widths will be the widths of the rectangles and height of the rectangles will be the Frequency Density. Fd = class width / frequency. The area of the rectangles will represent frequency since frequency = Fd x class width. The height and width of the rectangles can be in different ratios. The Fd values can be times by 10 and the class widths can be times by 2 before the Histogram is drawn.
Cumulative Frequency Graphs
You draw cumulative frequency graphs from frequency tables. You use the upper limit of the class widths as the x-coordinate, and you use the cumulative frequency as the y-coordinates for the points that you plot. You need connect the points with a smooth curve. The lower limit of the first class width will be where the graph starts from on the x-axis.
Frequency Polygons
You draw frequency polygons from frequency tables. You use the middle value of the class widths as the x-coordinate, and you use the frequency as the y-coordinates for the points that you plot. You need connect the points with straight lines. The frequency polygon will start from the first point and end on the last point plotted.
