Stat Prof Ed Nov. 25 2023 MCT
Stat Prof Ed Nov. 25 2023 MCT
Stat Prof Ed Nov. 25 2023 MCT
A measure of central tendency is the value around which the data tend to
be centered. This is to further describe a set of data when the observations are
arranged according to a natural order (like from lowest to the highest).
Though tabular and graphical summaries convey general impressions on the data,
measures of central tendency or location give information about a single value in
which the data set of observations tends to cluster to. The three primary measures
f central tendency are the mean, the median, and the mode.
Mean
A. Arithmetic mean
Arithmetic mean is the average value of the data; it is obtained by adding the
scores and dividing the sum by the number of scores. Most often, this particular
average is referred to simply the mean. To represent the mean of a sample, we use
x (read as “x bar”), and to represent the mean of a population, we use the Greek
letter mu, μ (read as “mew”)
n
∑ xi
Sample mean ( x )= i=1
N
where is the ith observation i = 1, 2, . . . . n
n
Population
∑ xi
mean ( µ )= i =1
N
where is the ith observation i = 1, 2, . . . . N
1: Find the average of the scores: 80, 82, 76, 78, 82,
and 91
Solution
x=
∑ X i = 80+82+76+ 78+82+91 =81.5
N 6
Answer: The average score is 81.5
B. Weighted mean
This value is obtained by summing up the product of each score by its
corresponding weight, divided by the sum of the weights.
x w=
∑ Xi W i where is the ith observation; is the weight of
∑Wi
the ith Observation and is the sum of the
corresponding weights.
2. Renan has the following grades and the equivalent credit units
for each grade. Determine his GWA (grade weighted average)
Solution: x w =
∑ Xi W i =
∑ Xi f i =
12 ( 2 ) +13 ( 13 ) +14 (27 ) +15 ( 4 )+ 16 ( 3 ) +17(1)
=13.92
∑Wi ∑ fi 50
Answer: the average age of the students enrolled in statistics is 14.
D. Geometric Mean
The geometric mean of a set of n scores is the nth root of their product.
= √ 2 ( 3 )( 6 ) (7 ) (7 ) ( 8 ) ( 9 ) ( 9 ) ( 9 ) (10 ) = 6.327
10
x̅ geometric
Note: Geometric mean is usually used with such data as rates of change, ratios,
economic index numbers, and population sizes over consecutive time periods.
When applied to rate of returns, geometric mean is the constant rate of
change (or rate of return) that yields the same wealth at the end of several time
periods as do actual returns (Bowerman and O’Connell 2007). The geometric mean
must be used when working with percentages (which are derived from values). The
main benefit to using the geometric mean is that the actual amounts invested do
not need to be known; the calculation focuses entirely on the return figures
themselves.
Characteristics of the mean
1. The mean is affected by all the values in the distribution; thus, a change in any
of the scores will cause a change in the mean.
2. The mean is sensitive to extreme scores (known as outlier values). Suppose in
example 1 above, a score of 45 is added to the given 6 scores (whose average is
81.5), then the mean becomes 76.29 which is very far from 81.5. In this case,
45 is referred to as an outlier value.
3. The sum of the deviations about the mean is equal to zero:
Σ(x i−x̅ ) = 0. This is shown in table 4.1
Table 4.1
Table 4.2
Illustration of different sums of squared deviations
______________________from the mean__________
xi Σ(x i−x̅ ) Σ( x i−x ) 2 Σ(x i−75) 2 Σ(x i−90) 2
As shown in Table 4.2 the lowest sum is that of the square of the differences from
the mean (81.5) columns 4 and 5 show squared differences from 75 and from 90,
respectively.
Median (M d )
The median is a positional average. When the scores are ranked, the median
is the point where half is greater and half is lesser. The median of a set of scores is
the middle value when the scores are arranged in order of increasing magnitude.
After arranging the original scores in increasing (or decreasing) order, the median
will be either of the following:
a. If the numbers of scores is odd, the median is the centermost score.
b. If the numbers of scores is even, the median is found by computing the
average of the two middle numbers.
8: Find the median of the scores 7, 2, 3, 7, 6, 9, 10, 8, 9, 9, 10.
Solution: arrange the scores in increasing magnitude or ascending order
2, 3, 6, 7, 7, 8, 9, 9, 9, 10, 10
With these eleven scores, the number 8 is located in the exact middle, so 8 is
the median.
9: Find the median of the scores 7, 2, 3, 7, 6, 9, 10, 8, 9, 9
Solution: Again, arrange the scores
2, 3, 6, 7, 7, 8, 9, 9, 9, 10
The two centermost scores are 7 and 8. So, we find the mean of these two
scores.
7+8
=7.5 Thus, 7.5 is the median of the given scores.
2
1
Formula : Md= ( N +1 ) th item
2
The Mode, M °
The most frequently occurring value in the set of scores in a set of data is
referred to as the mode; it is a normal average.
Midrange
Another measure of central tendency, but not often used is the midrange.
The midrange is that average obtained by adding the highest score to the lowest
score and the n dividing the sum by 2.
13. The mean salary for a small company that pays monthly
salaries to its employees as shown in the frequency distribution
Salaries frequency xf
7,000 8 56,000 mean salary is 9, 488
8,000 11 88,000
9250 14 129,000
10,500 9 94,500
17,000 2 34,000
25,000 1 25,000
TOTAL 45 427,000
2. Given the frequency distribution table below, compute for the measures of
central tendency.
C.I Cf
53 – 58 3
47 – 52 4
41 – 46 1
35 – 40 2
29 – 34 10
23 – 28 11
17 – 22 4
11 – 16 3
5 – 10 2______
N = 40
3. A study was conducted to determine the level of awareness of the residents of
a certain municipality on the causes of hypertension. The accompanying table
shows the result with respect to the gender of the respondents.