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Facultad de Comercio, Turismo y Ciencias Sociales Jovellanos
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Review: everyday use of percentages
Frequency tables
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What do you have to know?
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Review: everyday use of percentages
I (^) Percentages are an important part of our everyday lives. Examples: I (^) Shops advertise discounts on products. These discounts are percentages I (^) Financial institutions quote interest paid for money invested as a percentage. I (^) A salesperson may be given a percentage of the sales made (commission) as payment for selling goods. I (^) When you say that a is k% of b, you mean that:
a =
k 100
· b.
Examples: I (^) 5 is 5% of 100: 5 = 1005 · 100 I (^) 20 is 10% of 200: 20 = 10010 · 200
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Notation and nomenclature. Frequencies and percentages
I (^) Sample size: n. I (^) Values of a statistical variable in a sample: xi , i = 1,... , k. (We denote by k the number of different values of the variable in the sample). I (^) Frequency.- The number of times that a particular category/value xi appears in a sample (Denoted ni ). I (^) Relative frequency.- The fraction of times that a particular category/value xi appears in the sample. To find the relative frequencies, divide each frequency (ni ) by the total number of individuals in the sample (n). Relative frequencies can be written as fractions, percent, or decimals. When expressed as percentages, they will be denoted by pi.
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Frequencies and percentages: example
(Source: J. Buglear, Stats to go, Butterworth-Heinemann, Oxford,
- ) The number of passengers in each of 20 cars boarding a ferry were: 2 3 5 2 4 2 4 4 4 3 2 5 3 2 3 4 4 3 2 4 I (^) n = 20 I (^) k = 4 x 1 = 2, x 2 = 3, x 3 = 4, x 4 = 5 I (^) n 1 = 6, n 2 = 5, n 3 = 7, n 4 = 2 I (^) p 1 = 30%, p 2 = 25%, p 3 = 35%, p 4 = 10%
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Frequency tables (tablas de frecuencias)
I (^) Frequency table.- A table listing the frequency (number) and/or the percentage of observations in different categories or ranges. I (^) Frequency table in the example of ferry passengers:
xi ni pi 2 6 30% 3 5 25% 4 7 35% 5 2 10%
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Cumulative frequencies and percentages
(frecuencias y porcentajes acumulados)
Only for ordinal and numerical statistical variables: I (^) Cumulative frequency.- The sum of the frequencies of all the values up to a given value. If the values (or ordinal categories) x 1 <... < xk appear with respective frequencies n 1 ,... , nk , then the cumulative frequency of xi is equal to n 1 +... + ni. It is denoted Ni. I (^) Cumulative percent.- The sum of the percentages of all the values up to a given value. If the values (or ordinal categories) x 1 <... < xk appear with respective percents p 1 ,... , pk , then the cumulative frequency of vi is equal to p 1 +... + pi. It is denoted Pi.
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Cumulative frequencies and percentages: example of
passengers
xi ni pi 2 6 30% 3 5 25% 4 7 35% 5 2 10%
N 1 = 6 P 1 = 206 · 100% = 30% N 2 = 6 + 5 = 11 P 2 = 1120 · 100% = 55% N 3 = 6 + 5 + 7 = 18 P 3 = 1820 · 100% = 90% N 4 = 6 + 5 + 2 = 20 = n P 4 = 2020 · 100% = 100%
Remark: Nk = n 1 +... + nk = n and Pk = p 1 +... + pk = 100%
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Example of a pie chart
Pie chart - smoking habits
daily smokers occasionally smokers non smokers
Pie chart associated to the frequency table:
xi ni daily smoker 30 occas. smoker 24 non smoker 96
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Pareto chart
I (^) Named after Vilfredo Pareto. I (^) Valid: all types of variables (categorical, ordinal or numerical). I (^) Recommended for variables with few different values in the sample (k), specially when the values indicate “causes”. I (^) Description: I (^) It contains both bars and a line graph. I (^) Values are represented in descending order by bars. I (^) The cumulative percentages are represented by the line. I (^) Each rectangle corresponds to a possible value or category of the variable. I (^) The left vertical axis is the frequency of occurrence. I (^) The right vertical axis is the cumulative percentage of the total number of occurrences.
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Example of a Pareto chart
(Source: whatis.techtarget.com)
The business manager first thought about customer dissatisfaction with the clothing line he was selling. After analyzing a customer survey, however, he realized that parking difficulties, rude sales people and poor lighting were hurting his business most.
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Introduction to histograms: grouped data
I (^) Sometimes, the number of different values of the statistical variable is very high (weight of people measured in kg., length of people measured in cm., for instance). I (^) Before charting data, it is useful to previously group them into class intervals. I (^) Notation: I (^) Class intervals will be denoted by I 1 ,... , Ik. I (^) The endpoints of interval Ii will be denoted respectively by li− 1 and li. I (^) Frequency of interval Ii : ni. I (^) Percentage of individuals falling in interval Ii : pi.
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Histogram
I (^) Only valid for quantitative statistical variables. I (^) Recommended for variables with many different values in the sample, previously grouped into intervals. I (^) Description: I (^) A set of rectangles sitting on the horizontal axis. I (^) The bases of the rectangles are the class intervals I (^) The heights of the rectangles are equal to: I (^) their frequencies (frequency histogram) or to I (^) their percentages (percentage histogram).
(Another variant is the so-called density histogram. It is similar to frequency histogram except heights of rectangles are calculated by dividing relative frequencies by class width.)
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Example of a histogram
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Statistical variables expressed as functions of other
statistical variables: example
I (^) A company awards employees a bonus payment of 100 Eur (monthly) and an additional payment of 40 Eur for each child. I (^) Frequency table of X = “number of children of the employee”: xi ni pi 0 8 50% 1 4 25% 2 3 18.75% 3 1 6.25% I (^) If Y =“received bonus payment”, then Y = 100 + 40X. I (^) Frequency table of Y : xi ni pi 100 8 50% 140 4 25% 180 3 18.75% 220 1 6.25%
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What do you have to know?
I (^) Answering questions like: What is k% of b? a is what percent of b? a is k% of what? I (^) The meaning of the terms frequency, cumulative frequency, percentage and cumulative percentage. I (^) Calculate percentages from frequencies. I (^) Calculate frequencies from percentages (known sample size). I (^) Calculate cumulative frequencies from frequencies. I (^) Calculate the number and percentage of the individuals in a sample satisfying a certain condition. I (^) Calculate the sample size from the frequency table. I (^) Construct frequency tables from raw datasets. I (^) Select an appropriate chart for each kind of data. I (^) Determine the complete frequency table from any kind of chart. I (^) Understand and interpret the information provided by charts. I (^) Express a statistical variable Y as a function of another variable X. I (^) Understand the meaning of the sum of values of a quantitative variable in a sample. Calculate it.
