Section 1 : Central Tendency
 

[mean][mode][median]

 

 

 

The Mean - symbol μ (mu)

This is the average value for a set of single data values. It is calculated by adding up all the values and dividing by the number of values (n).

The Mean for grouped data

This is similar to some extent with the median for grouped data. No exact value can be found. It can only be estimated.

method:

find the mid-value for each group of data
(call these values x 1...x2...x3 etc.)

in turn, multiply the mid-value for each group by its frequency ( f1x1 ... f2x2...etc.)

sum the (product of mid-value x frequency)
(f1x1 + f2x2 + f3x3 ... etc.)

divide the sum by the sum of all frequencies,

mean for grouped data

note that Σ f = n

Example

The table represents the heights of pupils within a school.

Using the data calculate the mean value.

class

(height - h) cm

mid value

(x)

frequency

(f)

mid value x frequency

(fm)

70ineq symbol h < 75

72.5

2

145

75 ineq symbol h <80

77.5

7

542.5

85 ineq symbol h <90

87.5

21

1837.5

95 ineq symbol h <100

97.5

15

1462.5

105ineq symbol h <110

107.5

12

1290

 

total no. results

57

5277.5

estimated mean - equation #2

 

back to top

 

The Mode & Modal Class

For single values in a set of data, the mode is simply the value with the highest frequency.

For grouped values in a set of data, the modal class is simply that class/group with the highest frequency.

 

back to top

 

The Median

The median is the middle value of a set of data.

1__3__4__5__9__10__21__ 32__45

The median is therefore 9

The Median for 'grouped values'

For grouped values finding the median is more difficult. It cannot be found exactly, but is estimated using interpolation, which is essentially an educated guess.

The method is:

find which result is in the middle by dividing the total number of results in half. For example if there are 91 results , the median is result number 46(45.5 rounded up).

identify the modal class(the group of values/results) that contains the median

count how many numbers there are in this group

identify the frequency for the group

find the order of the median within the group

the median can be calculated as being the first number in the group plus a proportion of the number of results in the group

median grouped data

Example

The table represents the heights of pupils within a school.

Using the data calculate the median value.

class

(height - h) cm

class width

 

frequency

(f)

70ineq symbol h < 75

5

2

75 ineq symbol h <80

5

7

85 ineq symbol h <90

5

21

95 ineq symbol h <100

5

15

105ineq symbol h <110

5

12

 

total no. results

57

Since there are 57 values/results, the median occurs at position 28.5, ie. 29th to the nearest whole number.

This means that the median is the 29th value.

There are 2 values in the region 70 ineq symbol h < 75.

and a further 7 values in the region 75 ineq symbol h <80.

The so the median lies at the 20th value within the range 85 ineq symbol h <90 . This group has 21 values in it.

note: (9 + 20 = 29)

There are 5 numbers in the group.

The first number in the group is 85.

Using our equation:

median grouped data

median grouped data

median = 85 + 4.76 = 89.76 ~ 90(rounded)

 

back to top

 

your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]

VIDEO

the mean
the median
stand. deviation 1
stand. deviation 2
stand. deviation 3
z-scores
confidence interval
goodness of fit
distrib. sample mean
t interval
chi-squared test
 
more...Video Library
 

INTERACTIVE

normal distribution
mean,median comprd 1
mean,median comprd 2
type I & II errors
linear regression
histogram,box whisker
 
 

EXAM PAPERS(.pdf)

Edxl S1 Statistics spec.
Edxl S1 Statistics ans.
Edxl S2 Statistics spec.
Edxl S2 Statistics ans.
Edxl S3 Statistics spec.
Edxl S3 Statistics ans.
 

TOPIC NOTES(.pdf)

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google