In this section we recap the statistical measures mean, median, mode and range. The mean, median and mode give an indication of the 'average' value of a set of data, i.e. some idea of a typical value. The range, however, provides information on how spread out the data is, i.e. how varied it is.
Mean  = 
For 1, 2, 2, 3, 4
Mean  =  
=  
=  2.4 
For 1, 2, 2, 3, 4
Mode = 2
For 1, 2, 2, 3, 4, 4, 5
Mode = 2 and 4
Median  =  middle value when data is arranged in order 
For 1, 2, 2, 3, 4
Median  =  2 
For 1, 2, 2, 3, 4, 4
Median  =  
=  2.5 
Range  =  largest value – smallest value 
For 1, 2, 2, 3, 4
Range  =  4 – 1 
=  3 
In this section, we extend these basic ideas to grouped data.
The shoe sizes for a class are summarised in the table shown.
Calculate:
Shoe Size  Frequency 

4  2 
5  4 
6  7 
7  5 
8  6 
9  3 
10  3 
the mode,
the median
So the median  =  =  7 
the mean
(x) Size  (f) Frequency  (f x) Frequency × Size 
4  2  2 × 4 = 8 
5  4  4 × 5 = 20 
6  7  7 × 6 = 42 
7  5  5 × 7 = 35 
8  6  6 × 8 = 48 
9  3  3 × 9 = 27 
10  3  3 × 10 = 30 
Total  30  210 
The mean  =  =  =  7 
the range
The range  =  highest value – lowest value 
=  10 – 4  
=  6 
n + 1  th 
2 
157 + 1  th 
2 
30 + 1  th 
2 
The table shows the Morse code for 26 letters and how long it takes to send each letter.
If a letter is frequent we want to be able to send it quickly. The following table shows the 6 most frequent letters in 4 languages:
Complete the following table of the mean, median and modal sending times for the 6 most frequent letters in each language.
English :  mean time  = 
 
= 
 
median time : 
 
median  = 

French :  mean time  = 
 
= 
 
median time : 
 
median  = 

Italian :  times are 1, 11, 5, 3, 5, 7, so modal time is 5. 
Spanish :  times are 1, 5, 11, 5, 7, 3, so modal time is 5. 
Use your table in part (a) to decide which two languages are likely to send the quickest messages in Morse. Explain how you decided.
Samuel Morse invented the code. Messages in his own language are quick to send. Look at the table of the 6 most frequent letters in each language.
Which one of these letters has a code which suggests that Samuel Morse's own language was English? Explain how you decided.
Data on the number of minutes that a particular train service was late have been summarised in the table. (Times are given to the nearest minute.)
Minutes Late  Frequency 

on time  19 
15  12 
610  9 
1120  4 
2140  4 
4160  2 
over 60  0 
How many journeys have been included?
What is the modal group?
Estimate the mean number of minutes the train is late for these journeys.
Minutes Late  Midpoint (x)  Frequency (f)  (f x) 
On time  0  19  0 
15  3  12  36 
610  8  9  72 
1120  15.5  4  62 
2140  30.5  4  122 
4160  50.5  2  101 
Total  50  393 
Mean value  ≈  ≈  7.86 minutes 
Which of the two averages, mode and mean, would the train company like to use in advertising its service? Why does this give a false impression of the likelihood of being late?
Estimate the probability of a train being more than 20 minutes late on this service.
Estimate  =  =  0.12  =  12% 