STAT 245 Lecture Notes - Lecture 3: Box Plot, Unimodality, Standard Deviation
Stat 245 Lecture 3 – Describing Data with Numerical Measures
1. The Mean (or Average)
Symbols: Population mean,
Sample mean,
Formula:
Let denote the observation in the data set. Thus,
Population mean,
Sample mean,
Note: Sometimes, we just simplify the notation to simply .
Example: We are given the heights in inches of a simple random sample of 25 women:
58.2, 59.5, 60.7, 60.9, 61.9, 61.9, 62.2, 62.2, 62.4, 62.9, 63.1, 63.9, 63.9, 64.0, 64.5, 64.1, 64.8, 65.2, 65.7,
66.2, 66.7, 67.1, 67.8, 68.9 and 69.6
The sample mean height of the 25 women is
2. The Median
Symbol: M
Note: The first step in determining the median is to rearrange the data in ascending order, i.e. from
smallest to largest. The median is the middle-most value when the data values are arranged in ascending
order.
Location Formula:
Note: The above formula is NOT for the value of the median. It is for the position of the median.
Example 1: Let us consider the heights in inches of the 25 women in the previous example.
58.2, 59.5, 60.7, 60.9, 61.9, 61.9, 62.2, 62.2, 62.4, 62.9, 63.1, 63.9, 63.9, 64.0, 64.5, 64.1, 64.8, 65.2, 65.7,
66.2, 66.7, 67.1, 67.8, 68.9 and 69.6
What is the median of this data set?
Solution:
To find the median, the first step is to rearrange the data in ascending order, i.e. from smallest to largest.
Position
1
2
3
4
5
6
7
8
9
10
11
12
13
Height
58.2
59.5
60.7
60.9
61.9
61.9
62.2
62.2
62.4
62.9
63.1
63.9
63.9
Position
14
15
16
17
18
19
20
21
22
23
24
25
Height
64.0
64.1
64.5
64.8
65.2
65.7
66.2
66.7
67.1
67.8
68.9
69.6
The sample size is observations.
The median is located at the
position.
And the value at the 13th position, i.e. the median is inches.
Example 2: A student reported her 10 grades,
77, 86, 58, 67, 75, 77, 71, 65, 77 and 92
What is the median of this data set?
Solution:
The first step is to rearrange the data in ascending order:
Position
1
2
3
4
5
6
7
8
9
10
Grade
58
65
67
71
75
77
77
77
86
92
The sample size is observations.
The median is located at the
position.
Therefore, we take the value at the
position to be exactly halfway between the values of 75 and 77,
or the midpoint value of 75 and 77.
Thus, the median of the above data set is
5th position 6th position
3. The Mode
Symbol: m
The mode of a variable is the value that occurs most often in the data set.
It is a good idea to rearrange the data in order so that we can look for values that occur most often.
Example 1: Let us look at the 10 grades shown in the previous example:
77, 86, 58, 67, 75, 77, 71, 65, 77 and 92
What is the mode of the data set?
Solution:
The first step is to rearrange the data in order:
Position
1
2
3
4
5
6
7
8
9
10
Grade
58
65
67
71
75
77
77
77
86
92
We can see that the value that occurs most often is 77.
Hence, the mode of the data set is
It is possible to have more than one mode in some data sets.
Example 2: Let us consider the heights in inches of the 25 women in a previous example.
58.2, 59.5, 60.7, 60.9, 61.9, 61.9, 62.2, 62.2, 62.4, 62.9, 63.1, 63.9, 63.9, 64.0, 64.5, 64.1, 64.8, 65.2, 65.7,
66.2, 66.7, 67.1, 67.8, 68.9 and 69.6
What is the mode of this data set?
Solution: The first step is to rearrange the data set in order:
Position
1
2
3
4
5
6
7
8
9
10
11
12
13
Height
58.2
59.5
60.7
60.9
61.9
61.9
62.2
62.2
62.4
62.9
63.1
63.9
63.9
Position
14
15
16
17
18
19
20
21
22
23
24
25
Height
64.0
64.1
64.5
64.8
65.2
65.7
66.2
66.7
67.1
67.8
68.9
69.6
We can see that there are three values that occur twice (this data set’s the highest number of occurrences)
, i.e. 61.9, 62.2 and 63.9.
Therefore, there are three modes in this data set,
, and
Occurs most often
Occurs twice Occurs twice Occurs twice
Document Summary
Stat 245 lecture 3 describing data with numerical measures. Sample mean, (cid:1839)(cid:1857)(cid:1866)= (cid:3020)(cid:3048)(cid:3040) (cid:3042)(cid:3033) (cid:3042)(cid:3029)(cid:3046)(cid:3032)(cid:3045)(cid:3049)(cid:3028)(cid:3047)(cid:3042)(cid:3041)(cid:3046) (cid:3048)(cid:3040)(cid:3029)(cid:3032)(cid:3045) (cid:3042)(cid:3033) (cid:3042)(cid:3029)(cid:3046)(cid:3032)(cid:3045)(cid:3049)(cid:3028)(cid:3047)(cid:3042)(cid:3041)(cid:3046) Let denote the (cid:3047) observation in the data set. Note: sometimes, we just simplify the notation to simply . We are given the heights in inches of a simple random sample of 25 women: The sample mean height of the 25 women is. Note: the first step in determining the median is to rearrange the data in ascending order, i. e. from smallest to largest. The median is the middle-most value when the data values are arranged in ascending order. Note: the above formula is not for the value of the median. It is for the position of the median. Let us consider the heights in inches of the 25 women in the previous example. To find the median, the first step is to rearrange the data in ascending order, i. e. from smallest to largest.
