Understanding Data Variables and Quartiles in Statistics
In this educational content, delve into the concepts of variables in statistics, distinguishing between qualitative and quantitative types, including discrete and continuous variables. Explore quartiles, their calculation for large datasets, and quickfire practice scenarios. Additionally, learn about notation for quartiles and percentiles, as well as grouped frequency data and estimation of mean values. Engage with practical examples and explanations to enhance your statistical knowledge.
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S1: Chapters 2-3 Data: Location and Spread Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 5th September 2014
Types of variables ? In statistics, we can use a variable to represent some quantity, e.g. height, age. This could be qualitative (e.g. favourite colour) or quantitative (i.e. numerical). Variables are often used differently in statistics than they are in algebra. ? In statistics, this would mean: Sum over the values of the variable we re collected (i.e. our data). 2 types of variable: Discrete variables Continuous variables Has specific values. e.g. Shoe size, colour, website visits in an hour period, number of siblings, Can have any value in a range. e.g. Height, distance, weight, time, wavelength, ? ?
Quartiles for large numbers of items Rule: Find 1 If not whole, round up. If whole, use this item and one after. 4 or 1 2 or 3 4 of ?. Then: What item do we use for each quartile when ? = LQ Median UQ 8th ? 16th ? 24th ? 31 15th ? 10th ? 5th ? 19 5th ? 3rd and 4th ? 2nd ? 6 11th ? 14 4th ? 7th and 8th ? Under what circumstances do we not round? When we have a grouped frequency table involving a continuous variable. ?
Quickfire Quartiles LQ Median UQ 1, 2, 3 1 ? 2 ? 3 ? 1.5 ? 2.5 ? 3.5 ? 1, 2, 3, 4 1, 2, 3, 4, 5 1.5 ? 2 ? 4.5 ? 2 ? 3.5 ? 5 ? 1, 2, 3, 4, 5, 6
Notation for quartiles/percentiles ?1 Lower Quartile: ? ?2 Median: ? ?3 Upper Quartile: ? ?57 57th Percentile: ?
Grouped Frequency Data Recap ? This type of data is continuous. Height ? of bear (in metres) Frequency 0 < 0.5 4 0.5 < 1.2 20 1.2 < 1.5 5 1.5 < 2.5 11 ?? ?=46.75 ? Estimate of Mean: ? = = 1.17? 40 ? ? The midpoints of each interval. They re effectively a sensible single value used to represent each interval. ? What does the variable ? represent? It s the sample mean of ?. It indicates that our mean is just based on a sample, rather than the whole population. Why the bar (horizontal line) over the ?? ? Why is our mean just an estimate? Because we don t know the exact heights within each group. Grouping data loses information. ?
Grouped Frequency Data Recap Height ? of bear (in metres) Frequency 0 < 0.5 4 0.5 < 1.2 20 1.2 < 1.5 5 1.5 < 2.5 11 0.5 < 1.2 ( modal means most ) Modal class interval: ? There are 40 items, so determine where 20th item is. ? Median class interval: 0.5 < 1.2
Using STATS mode on your calculator Height ? of bear (in metres) Frequency 0 < 0.5 4 Work out the mean for this example first using proper workings. 0.5 < 1.2 20 1.2 < 1.5 5 1.5 < 2.5 11 Go to SETUP (SHIFT MODE). Press down for the second page of menu, and select STAT. You want Frequency ON . (Note that you won t have to do this again in future) MODE STAT Select 1-VAR (as there is only 1 variable here frequency is not a variable!) Enter your x values, pressing = after each one. Navigate to the top of your table to enter your frequencies. Press AC to bank your table. SHIFT 1 for STAT . Select each Sum or Var . Once you ve selected a statistic to use, it ll appear in your calculation. Once you want to calculate the value, press =. Try entering ? ?. (For this example: 1.16875) MODE COMP to go back to normal computation mode. Important note: Confusingly, your calculator means ?? when you enter ?. And ? = ?, i.e. it s interpreting the data as if it was listed out with duplicated. 1. 2. 3. 4. 5. 6. 7. Warning: You still need to show working in the exam.
Whats different about the intervals here? Weight of cat to nearest kg Frequency 10 12 7 13 15 2 16 18 9 19 20 4 There are GAPS between intervals! What interval does this actually represent? 10 12 9.5 12.5 ? Lower class boundary Upper class boundary ? Class width = 3
Identify the class width Time ? taken (in seconds) Distance ? travelled (in m) 0 3 0 d < 150 ? ? 150 d < 200 7 11 ??? ? < ??? Lower class boundary = 3.5 ? Lower class boundary = 200 ? Class width = 3? Class width = 10? Speed ? (in mph) Weight ? in kg 10 s < 20 10 20 20 ? < 29 21 30 ?? ? < ?? ?? ?? Lower class boundary = 29? Lower class boundary = 30.5 ? Class width = 2 ? Class width = 10?
S2 Chapters 2/3 Interpolation
RECAP: Quartiles of Frequency Table Age of squirrel Frequency Cumulative Freq 1 5 5 2 8 13 3 11 24 4 5 29 29 squirrels. 29 So look at 8th squirrel. Occurs within second group, so ?1= 2 4= 7.25 ?1? ? 29 2= 14.5 so use 15th squirrel. Occurs in third group, so ?2= 3 ?2? ? 3 4 29 = 21.75 so use 22nd squirrel. Still in third group, so ?3= 3 ?3? ?
Estimating the median GCSE Question ? Answer = 13.5 + 8 = 21.5
Estimating the median At GCSE, you were only required to give the median class interval when dealing with grouped data. Now, we want to estimate a value within that class interval. Weight of cat to nearest kg Frequency 10 12 7 13 15 2 16 18 9 19 20 4 (Why not the 11.5 item?) Frequency up until this interval Frequency at end of this interval Item number we re interested in. 18 ? ? 11 ? 9 ? ? 15.5kg ? 18.5kg Weight at start of interval. Weight at end of interval. 2 9 3 = 16.17?? ? Median= 15.5 +
Estimating other values Weight of cat to nearest kg Frequency 10 12 7 13 15 2 16 18 9 19 20 4 5.5 7 3 = 11.86?? ? 7.5 9 3 = 18?? ? = 9.5 + LQ = 15.5 + UQ 0.48 2 3 = 13.22?? ? 34th Percentile = 12.5 +
You should have a sheet in front of you 1 29 500 = 1017.74 years 1000.5 + ? 1a 26 29 500 = 1448.78 years ? 1000.5 + 1b 10 35 300 = 1786.21 years ? 1700.5 + 1c Interquartile Range: 1786.21 1017.74 = 768.47 years ? 1d 5.2 17 60 = 58.35cm ? 40 + 2a 6.8 8 300 = 555cm ? 300 + 2b ? 555 58.35 = 496.65cm 2c
Exercises Page 34 Exercise 3A Q4, 5, 6 Page 36 Exercise 3B Q1, 3, 5
S2 Chapters 2/3 Variance and Standard Deviation
What is variance? Distribution of IQs in L6Ms5 Distribution of IQs in L6Ms4 ????????? ????????? ?? ?? 110 110 Here are the distribution of IQs in two classes. What s the same, and what s different?
Variance Variance is how spread out data is. Variance, by definition, is the average squared distance from the mean. 2 ? ? ? ?2= Distance from mean Squared distance from mean Average squared distance from mean
Simpler formula for variance Variance The mean of the squares minus the square of the mean ( msmsm ) 2 ???????? = ?2 ? ? ? ? ? Standard Deviation ? = ???????? The standard deviation can roughly be thought of as the average distance from the mean.
Starter Calculate the variance and standard deviation of the following heights: 2cm 3cm 3cm 5cm 7cm Variance = 19.2 42= 3.2cm ? Standard Deviation = 3.2 = 1.79cm ?
Practice Find the variance and standard deviation of the following sets of data. 2 4 6 ? ? Variance = 2.67 Standard Deviation = 1.63 1 2 3 4 5 ? Variance = 2 Standard Deviation = 1.41 ?
Extending to frequency/grouped frequency tables We can just mull over our mnemonic again: Variance: The mean of the squares minus the square of the means ( msmsm ) 2 ???????? = ??2 ?? ? ? ? ? Bro Tip: It s better to try and memorise the mnemonic than the formula itself you ll understand what s going on better, and the mnemonic will be applicable when we come onto random variables in Chapter 8.
Example Height ? of bear (in metres) Frequency 0 < 0.5 4 0.5 < 1.2 20 1.2 < 1.5 5 1.5 < 2.5 11 ??2= 67.81 ?? = 46.75 ? = 40 ? ? ? 2 ???????? =67.81 46.75 40 ? = 0.33 40
Sometimes were helpfully given summed data: Shoe Size ? Frequency 10 7 11 2 12 9 13 4 ??2= 2914 ?? = 252 ? = 22 2 ???????? =2914 252 22 = 1.25 ? 22
Exercises Page 40 Exercise 3C Q1, 2, 4, 6 Page 44 Exercise 3D Q1, 4, 5
Recap ?2= 50, ?2= 6 ? = 10, ? = 5 ? y2= 100, ?2= 4 y = 20, n = 5 ? ??2= 1000, ?? = 100, ?2= 0 ? = 10 ? ??2= 400, ?? = 20, ?2= 75 ? = 4 ?
S2 Chapters 2/3 Coding
Starter What do you reckon is the mean height of people in this room? Now, stand on your chair, as per the instructions below. INSTRUCTIONAL VIDEO Is there an easy way to recalculate the mean based on your new heights? And the variance of your heights?
Starter Suppose now after a bout of stretching you to your limits , you re now all 3 times your original height. What do you think happens to the standard deviation of your heights? It becomes 3 times larger (i.e. your heights are 3 times as spread out!) ? What do you think happens to the variance of your heights? It becomes 9 times larger ? (Can you prove the latter using the formula for variance?)
The point of coding Cost ? of diamond ring ( ) 1010 1020 1030 1040 1050 We code our variable using the following: ? =? 1000 10 New values ?: 1 2 3 4 5 ? ? Standard deviation of ? (??): ? therefore ? Standard deviation of ? (??): 10 2
Finding the new mean/variance Old mean ? New mean ? Old variance Coding New variance 16 ? 4 ? 36 4 ? = ? 20 36 ? 4 ? 72 16 ? = 2? 85 ? 36 ? 35 4 ? = 3? 20 ? =? 3 2 40 ? 6 ? 20 2 ? =? + 10 7 ? 3 ? 11 27 3 ? =? 100 300 ? 125 ? 40 5 5
Exercises Page 26 Exercise 2E Q3, 4 Page 47 Exercise 3E Q2, 3, 5, 7
Chapters 2-3 Summary I have a list of 30 heights in the class. What item do I use for: ? ? 8th Between 15th and 16th 23rd ? ?1? ?2? ?3? For the following grouped frequency table, calculate: Height ? of bear (in metres) Frequency 0 < 0.5 4 0.5 < 1.2 20 1.2 < 1.5 5 1.5 < 2.5 11 0.25 4 + 0.85 20 + 40 16 20 0.7 = 1.06? =46.75 40 ? = a) The estimate mean: = 1.17? ?? 3?? b) The estimate median: ? 0.5 + 2 ?2=67.8125 46.75 40 ? c) The estimate variance: (you re given ? 2= 67.8125) = 0.329 ?? 3?? 40
Chapters 2-3 Summary What is the standard deviation of the following lengths: 1cm, 2cm, 3cm ?2=14 3 22=2 2 3 ? 3 ? = The mean of a variable ? is 11 and the variance 4. The variable is coded using ? =?+10 3. What is: ? ? = ? ?? a) The mean of ?? b) The variance of ?? ?=? ? ? A variable ? is coded using ? = 4? 5. For this new variable ?, the mean is 15 and the standard deviation 8. What is: ? ? = ? a) The mean of the original data? b) The standard deviation of the original data? ??= ? ?
