Frequency-distribution-1.pdf

Frequency distribution

Frequency • Frequency is how often something occurs. Example: Sam played football on - Saturday morning - Saturday afternoon - Thursday afternoon

The frequency was 2 on Saturday, 1 on Thursday and 3 for the whole week.

Frequency distribution A frequency distribution is a tabulation or grouping of data into appropriate categories showing the number of observations in each group or category. Consider the given data below which show the numbers of newspapers sold at a local shop over the last 10 days. 22, 20, 18, 23, 20, 25, 22, 20, 18, 20 The numbers shown above are called raw data.

Papers Sold

Frequency

25

1

24

0

23

1

22

2

21

0

20

4

19

0

18

2

The total frequency in this distribution is (σ 𝑓 = 10)

Example: Shown below are the scores of 60 students in a 30-point Math Quiz.

11 16 21 11 19 19 16 16 26 11

19 16 11 19 13 15 23 24 15 15

11 15 13 24 18 11 10 12 11 18

15 17 21 12 20 25 17 21 14 12

16 10 10 22 21 29 11 13 10 20

10 27 16 13 11 23 27 12 12 13

Scores 29 28 27 26 25 24 23 22 21 20

f 1 0 2 1 1 2 2 1 3 2

Scores f 19 3 18 2 17 2 16 6 15 5 14 1 13 5 12 5 11 9 10 5 σ 𝑓 = 60

Parts of a grouped frequency table • Class Limits – groupings or categories defined by lower and upper limits. Example: 26-30 21-25 16-20 Lower class limits are the smallest numbers that belong to the different classes.

Upper class limits are the highest numbers that belong to the different classes.

Parts of a grouped frequency table •Class size – width of each class interval.

Lower Limit Upper Limit 21 25 16 20 Class size = 5

Parts of a grouped frequency table • Class boundaries – a point that represents the halfway point between two successive classes. Example: C.I. C.B L.L U.L L.C.B U.C.B 31 - 35 30.5 - 35.5 26 - 30 25.5 - 30.5 21 - 25 20.5 - 25.5 16 - 20 15.5 - 20.5

Parts of a grouped frequency table • Class marks – the midpoints of the lower and upper class limits. They can be found by adding the lower and upper limits and then dividing the answer by 2. Example: C.I Class mark (X) 31-35 33 26-30 28 21-25 23 16-20 18

STEPS IN CONSTRUCTING A grouped frequency table

1. Find the range of the values. Range = highest value – lowest value

STEPS IN CONSTRUCTING A grouped frequency table

2. Determine the class width by dividing the range by the desired number of groupings. The class size is the width of each class interval. 𝑟𝑎𝑛𝑔𝑒 𝐶= 1 + 3.222 log 𝑁

STEPS IN CONSTRUCTING A grouped frequency table 3. Set up the class limits of each class. 4. Tally the scores in the appropriate classes and then add the tallies for each class in order to obtain the frequency.

5. Set up the class boundaries. The class boundaries or true limits of a class is defined by a lower class boundary and an upper class boundary. 6. Solve the class mark or midpoint (x) of each class. This is obtained by adding the lower class limit and the upper class limit, then dividing it by 2.

Cumulative frequency distribution • The total frequency of all classes less than the upper class boundary of a given class is called the cumulative frequency of that class. A table showing the cumulative frequencies is called a cumulative frequency distribution. There are two types of cumulative frequency distributions. Less than cumulative frequency distribution (
Relative frequency distribution • The relative frequency of a class is the frequency divided by the total frequency or total number of observations and is generally expressed as a percentage.

𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑐𝑙𝑎𝑠𝑠 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

Graphical representation of the frequency distribution • Histogram – consists of a set of rectangles having bases on a horizontal axis which have the class boundaries as points, and centers on the class marks. The base widths correspond to the class size and the heights of the rectangles correspond to the class frequencies. • Frequency polygons – constructed by plotting class frequencies against class marks and connecting the consecutive points by a straight line.

Presenting Nominal Data Marital Status Single Married Widowed Separated

Frequency 25 10 5 20 N = 60

Presenting Nominal Data

•Pie Chart •Bar Graph

Presenting Ordinal Data Rank Excellent Very Good Good Fair Poor

Frequency 30 20 15 10 5 N = 80

Presenting Ordinal Data

•Bar Graph

Presenting Interval/Ratio Data

•Histogram •Frequency Polygon

Seatwork #3 Make a distribution table of the scores of 40 students in Accounting. 37 61 59 48 28 55 46 52

79 46 52 63 28 57 54 81

54 57 46 81 46 62 39 40

62 70 55 39 57 69 48 20

39 33 68 43 25 37 57 55

Measures of Central Tendency

• The mean (commonly called the average) of a set of n numbers is the sum of all numbers divided by n. The sum of the differences from the mean is equal to zero. • The median is the middle number when the number in a set of data is arranged in descending order. When there are even numbers of elements, the median is the mean of the two middle numbers. • The mode is the number that occurs most often in a set of data. A set of data can have more than one mode. If all the numbers appear the same number of times, there is no mode for that data set.

Mean of ungrouped data σ 𝑥𝑖 𝑋= 𝑁 where X = mean 𝑥𝑖 = data σ 𝑥𝑖 = sum of data N = total frequencies

Mean of ungrouped data Example: The following are the scores in Statistics quizzes during the first quarter.

70 72 77 86 78 84 79 Compute for the mean or the average of the scores

Median of ungrouped data •The arrangement of data in ascending or descending order is called an array. •The computation of the median of an ungrouped data needs an arrangement of the scores either in ascending or descending order. The median is the value of the middle score, that is, the value at the (𝑁 + 1 ÷ 2)𝑡ℎ position.

Median of ungrouped data Find the median of the following scores: a.) 12 35 48 50 55 60 65 b.) 105 107

120

111

101

115

Mode of ungrouped data If we look at a frequency distribution, we can easily determine the mode since it is merely the data with the largest frequency. Example: Find the mode of the following scores:

4

7

7

7

8

8

9

10 11 11 13

Weighted Mean

𝑤1 𝑥1 + 𝑤2 𝑥2 + ⋯ + 𝑤𝑛 𝑥𝑛 𝑥ҧ = 𝑤1 + 𝑤2 + ⋯ + 𝑤𝑛

Example Jillian wants to apply on Jose Rizal Honor Society; a society of scholars in First Asia Institute of Technology and Humanities (FAITH). In the said scholarship, each student should maintain a GPA of 85. Jillian wants to know if her grades can reach the GPA of scholars in FAITH. The table shows her grades.

Example SUBJECT GRADE UNIT Trigonometry 84 3 Komunikasyon sa 87 3 Akademikong Filipino Psychology 85 3 Philippine History 85 3 Intro to Computer 88 3

Mean of grouped data σ 𝑓𝑋𝑚 𝑥ҧ = 𝑁 where 𝑥ҧ = mean f = frequency 𝑋𝑚 = class mark σ 𝑓𝑋𝑚 = sum of the product of frequencies and class marks N = total frequencies

Example

Class Interval 63 – 70 55 – 62 47 – 54 39 – 46 31 – 38

Frequency 9 9 8 4 10

Median of grouped data 𝑁 − 𝑐𝑓𝑏 𝑥෤ = 𝑋𝐿𝐵 + 2 𝑖 𝑓𝑚 where 𝑥෤ = median 𝑋𝐿𝐵 = lower boundary of the median class N = total frequency 𝑐𝑓𝑏 = cumulative frequency before the median class 𝑓𝑚 = frequency of the median class i = size of class interval

Mode of grouped data • The first step is getting the modal class. The modal class is the class interval having the largest frequency. 𝑥ො = 𝑋𝐿𝐵 +

∆1 𝑖 ∆1 + ∆2

where 𝑋𝐿𝐵 = lower boundary of the modal class ∆1 = difference between the frequency of the modal class and the frequency preceding it ∆2 = difference between the frequency of the modal class and the frequency succeeding it i = size of the interval