Histograms
The
AREA of each bar represents the frequency
Sometimes a key like the red one in the
histogram below is used
We will now construct a histogram to
represent the following grouped continuous data. REMEMBER, discrete data is data
that can only take particular values (usually integers) like "the number of boys
in a class". Continuous data is data that can take any value at all like height.
HEIGHT OF PLANT 
FREQUENCY 
FREQUENCY DENSITY 
0≤x<2 
4 
2 
2≤x<4 
7 
3.5 
4≤x<8 
5 
1.25 
We need to work out the "frequency
density" in order to draw the bars correctly on the diagram, otherwise the bars
would be disproportionate when working with different widths. (Getting one
person between 1&2 meters tall is not the same as getting one person between
1&1.2 meters tall. The later 'deserves' a higher bar because the interval was
smaller.
Frequency density = ^{Frequency }/ _{Class width
}
Example: ^{7}/_{(42)} = 3.5 and ^{5}/_{(84)}
= 1.25 
This formula comes from the fact that the
area of each bar represents the frequency so height of bar ("frequency density")
times the width of the bar = the frequency.
*Just a note... When dealing with ages
remember that they always round down. When someone is 13½ they are not 14 but
13!*
Averages
Mean
Take the midpoint of each interval and
add them together
Divide by the sum of the frequencies
Example...
Value 
Frequency 
0≤x<6 
7 
6≤x<12 
2 
12≤x<18 
6 
Mode
The modal interval/class is the one with
the highest frequency density (the tallest bar)
Median
The middle value
It cuts histograms in half so that there
is an equal area on either side
The total number of plants sampled is 16
so the median height will be the 8th one along.
How tall is the 8th one along?...
There are 4 plants in the first bar so
this 8th one is NOT in the first bar.
The next bar contains the next 7 plants.
Therefore the 8th plant that we are looking for is the 4th plant out of these
seven. It therefore lies 4 sevenths of the way across this bar. This bar starts
at a plant height of 2 cm and goes up to 4cm. Therefore the plant we are
interested in lies 4 sevenths of the way between 2 and 4. This value is 2+(^{4}/_{7}
x 2) = 3 ^{1}/_{7}cm
