Averages and range
Section: Statistics | Syllabus: Cambridge IGCSE Mathematics (0580)
What Is the Mode?
- The mode is the value that appears the most often
- The mode of 1,2,2,5,6 is 2
- There can be more than one mode
- The modes of 1,2,2,5,5,6 are 2 and 5
- The mode can also be called the modal value
What Is the Median?
- The median is the middle value when you put values in size order
- The median of 4,2,3 can be found by ordering the numbers: 2,3,4 - and choosing the middle value, 3
- If you have an even number of values, find the midpoint of the middle two values
- The median of 1,2,3,4 is 2.5
- 2.5 is the midpoint of 2 and 3
- The midpoint is the sum of the two middle values divided by 2
What Is the Mean?
- The mean is the sum of the values divided by the number of values
- The mean of 1,2,6 is (1+2+6)3=3
- The mean can be a fraction or a decimal
- It may need rounding
- You do not need to force it to be a whole number - you can have a mean of 7.5 people, for example!
What Is the Range?
- The range measures how spread out the data is: range = largest value - smallest value
- The range of 3,7,9,15 is 15-3=12
- A larger range means the data is more spread out; a smaller range means the data is more consistent
- Unlike the mean, median and mode, the range is not itself an average - it describes spread, not a typical or central value
How Do I Know When to Use the Mode, Median or Mean?
- The mode, median and mean are different ways to measure an average
- In certain situations it is better to use one average over another
- For example, if the data has extreme values (outliers) like 1,1,4,50:
- The mode is 1
- The median is 2.5
- The mean is 14
- Don't use the mean - it's badly affected by extreme values
- If the data has more than one mode
- Don't use the mode - it is not clear which value to pick
- If the data is non-numerical, like dog, cat, cat, fish
- You can only use the mode
Finding Mean, Median, Mode and Range Together
Worked Example: Finding Mean, Median, Mode and Range
- Question: Find the mean, median, mode and range of this list of scores: 5,\,9,\,5,\,12,\,6,\,10,\,5,\,8,\,12.
- Step 1 (mean): Sum =5+9+5+12+6+10+5+8+12=72, and there are 9 values Mean=(72/9)=8
- Step 2 (median): Arrange in order: 5,5,5,6,8,9,10,12,12 - the middle (5th) value is 8
- Step 3 (mode): The value 5 appears three times, more than any other value, so the mode is 5
- Step 4 (range): 12-5=7
- Answer: mean =8, median =8, mode =5, range =7
Common Mistakes
MistakeFinding the median without first arranging the data in order
Fixalways sort the data from smallest to largest before picking out the middle value(s) - the median of unsorted data is meaningless
MistakeConfusing mode (the most frequent value) with median (the middle value) or reporting the frequency itself as the mode instead of the value
Fixthe mode is the data value that appears most often - not how many times it appears, and not the middle position
Mean, Median and Mode from a Frequency Table
- For data in a frequency table, Mean=(Σ(f× x)/Σ f) where x is each value and f is its frequency
- The median's position is found using the total frequency, then located using a running (cumulative) total down the table
- The mode is simply the value with the highest frequency
Worked Example: Averages from a Frequency Table
- Question: The table shows the number of goals scored by a football team in each of 20 matches.
Find the mean, median and mode number of goals.Goals, x 0 1 2 3 4 Frequency, f 3 6 7 3 1 - Step 1 (mean): Σ(f× x)=0(3)+1(6)+2(7)+3(3)+4(1)=0+6+14+9+4=33 Mean=(33/20)=1.65
- Step 2 (median): With 20 matches, the median is between the 10th and 11th values. Running totals: 3 (up to 0 goals), 9 (up to 1 goal), 16 (up to 2 goals) - so both the 10th and 11th values fall among the "2 goals" matches
- Step 3 (mode): The highest frequency is 7, for 2 goals, so the mode is 2
- Answer: mean =1.65, median =2, mode =2
Common Mistakes
MistakeDividing by the number of rows in the table (the number of different values) instead of the total frequency Σ f
Fixthe mean's denominator is always the total number of data items (sum of all the frequencies), not the number of distinct values listed
MistakeReporting the highest frequency itself (e.g. 7) as the mode, instead of the value it belongs to (2 goals)
Fixthe mode is the value (x) with the highest frequency, not the frequency number itself
Estimating the Mean from Grouped Data
Extended Only
- For grouped (continuous) data, the exact values are unknown, so the mean can only be estimated
- Use the midpoint of each class interval to represent every value in that class: midpoint=(lower bound+upper bound/2)
- Estimated mean =(Σ(f)/Σ f)
- The modal class is the class interval with the highest frequency (a whole interval, not a single value, since the exact values are unknown)
Worked Example: Estimating the Mean from Grouped Data
- Question: The table shows the heights, h cm, of 40 plants.
Calculate an estimate of the mean height, and state the modal class.Height, h (cm) 0 h<10 10 h<20 20 h<30 30 h<40 Frequency 6 14 15 5 - Step 1: Midpoints are 5,\,15,\,25,\,35
- Step 2: Σ(f)=6(5)+14(15)+15(25)+5(35)=30+210+375+175=790
- Step 3: Estimated mean=(790/40)=19.75 cm
- Step 4: The highest frequency (15) is for 20 h<30, so this is the modal class
- Answer: estimated mean =19.75 cm; modal class =20 h<30
Common Mistakes
MistakeUsing the lower or upper bound of a class interval instead of its midpoint when estimating the mean
Fixalways use the midpoint - it is the best single value to represent an entire class interval when the individual values are unknown
MistakeGiving the modal class as a single midpoint value instead of the full class interval
Fixthe modal class is stated as the whole interval (e.g. 20 h<30), since the exact values within a grouped class are not known
Working Backwards from Given Statistics
Extended Only
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