Mth 110 – Introduction to Statistics Session 4 – Measures of Dispersion
24/10/2011
MTH 110 – INTRODUCTION TO
STATISTICS
Session 4 – Measures of Dispersion, Variance and Standard Deviation Instructor: Manos Takas Email: m.takas@cityu.gr
Range Look at these two sets of data: 2, 3, 4, 5, 6 3,2,3,5,13 -3,2,3,5,13 They both have a mean of 4. However, you can see that the first data set (21 is more spread out than the second data set. The mean doesn't tell you this. _ To represent the data more accurately, you need the mean plus a measure of the spread or dispersion of the data. One simple measure of dispersion is the range. The range of a set of data is the highest value minus the lowest value
In this case the range of set [1] is 6 - 2 = 4. The range of set [2] is 13 (-3) = 16. The range is easy to calculate, but there are other measures of spread that are more useful.
1
24/10/2011
Variance and Standard Deviation The mean of the deviations from the mean squared is called the variance, usually written as σ 2
The variance is calculated as:
σ2
=
1 ∑ ( x − x )2 n
2
Sometimes it is easier to calculate a variance using the alternative formula:
σ
⎛∑x⎞ 1 = ∑ x2 − ⎜ ⎜ n ⎟ ⎟ n ⎝ ⎠
2
Standard Deviation The standard deviation σ is defined as the square root of the variance
⎛ ∑ x2 ⎞ 1 1 2 σ= ∑ x − ⎜ n ⎟ = n ∑ ( x − x )2 ⎜ ⎟ n ⎝ ⎠
2
Standard Deviation for Frequency Distributions
σ2 =
1 ∑ n
⎛ ∑ fx ⎞ fx 2 − ⎜ ⎜ n ⎟ ⎟ ⎝ ⎠
2
Where f represents the frequency of the x observation and n=Σf
2
24/10/2011
3
MTH 110 – INTRODUCTION TO
STATISTICS
Session 4 – Measures of Dispersion, Variance and Standard Deviation Instructor: Manos Takas Email: m.takas@cityu.gr
Range Look at these two sets of data: 2, 3, 4, 5, 6 3,2,3,5,13 -3,2,3,5,13 They both have a mean of 4. However, you can see that the first data set (21 is more spread out than the second data set. The mean doesn't tell you this. _ To represent the data more accurately, you need the mean plus a measure of the spread or dispersion of the data. One simple measure of dispersion is the range. The range of a set of data is the highest value minus the lowest value
In this case the range of set [1] is 6 - 2 = 4. The range of set [2] is 13 (-3) = 16. The range is easy to calculate, but there are other measures of spread that are more useful.
1
24/10/2011
Variance and Standard Deviation The mean of the deviations from the mean squared is called the variance, usually written as σ 2
The variance is calculated as:
σ2
=
1 ∑ ( x − x )2 n
2
Sometimes it is easier to calculate a variance using the alternative formula:
σ
⎛∑x⎞ 1 = ∑ x2 − ⎜ ⎜ n ⎟ ⎟ n ⎝ ⎠
2
Standard Deviation The standard deviation σ is defined as the square root of the variance
⎛ ∑ x2 ⎞ 1 1 2 σ= ∑ x − ⎜ n ⎟ = n ∑ ( x − x )2 ⎜ ⎟ n ⎝ ⎠
2
Standard Deviation for Frequency Distributions
σ2 =
1 ∑ n
⎛ ∑ fx ⎞ fx 2 − ⎜ ⎜ n ⎟ ⎟ ⎝ ⎠
2
Where f represents the frequency of the x observation and n=Σf
2
24/10/2011
3